Magnetic particle imaging devices and methods

ABSTRACT

A magnetic particle imaging device is provided. The device includes a magnetic field source configured to produce a magnetic field having a non-saturating magnetic field region, an excitation signal source configured to produce an excitation signal in the non-saturating magnetic field region that produces a detectable signal from magnetic particles in the non-saturating magnetic field region, and a signal processor configured to convert a detected signal into an image of the magnetic particles. Aspects of the present disclosure also include methods of imaging magnetic particles in a sample, and methods of producing an image of magnetic particles in a subject. The subject devices and methods find use in a variety of applications, such as medical imaging applications.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application is a continuation-in-part of U.S. patentapplication Ser. No. 12/737,214, filed Dec. 16, 2010, which is anational stage application of PCT Application No. PCT/US2009/003764,filed Jun. 23, 2009, which claims priority to U.S. Provisional PatentApplication No. 61/074,931, filed Jun. 23, 2008, the disclosures of eachof which are hereby incorporated by reference in their entirety. Thisapplication also claims the benefit of priority under 35 U.S.C. §119(e)to U.S. Provisional Patent Application Nos. 61/340,542, filed Mar. 17,2010, and 61/442,229, filed Feb. 12, 2011, the disclosures of each ofwhich are hereby incorporated by reference in their entirety.

INTRODUCTION

Tomography is a technique of imaging by sections or sectioning, throughthe use of any kind of penetrating wave, such as radio-frequency waves,sound waves, x-rays, gamma rays, electron-positron annihilation waves,etc. Tomography can be used to produce two-dimensional cross-sectionalimage slices of a sample or subject in the tomographic device. Theseslices can be superimposed to form a three-dimensional image of thesample or subject. The data acquired by the tomographic device isanalyzed by a mathematical procedure called tomographic reconstructionto produce the images. Tomographic reconstruction is typically performedusing computers (e.g., computed tomography).

Magnetic particle imaging (MPI) is a tomographic or volumetric imagingtechnique that directly detects the magnetization from magneticparticles. The basic principle of MPI involves applying a magnetic fieldto magnetic particles in a selected region (e.g., magnetic particlecontrast agents injected into the blood stream or labeled into or oncells) and detecting the magnetic fields generated by the magneticparticles. Similar to tomographic reconstruction, the data acquired frommagnetic particle imaging can be processed using algorithms to produceimages of the magnetic particles in the sample or subject. Similar tothe tomographic imaging techniques discussed above, MPI has potentialapplications in medicine, such as in medical imaging, e.g., heart andblood vessel imaging, cell tracking, interventional radiology, andcancer detection. For example, a tracer or contrast agent that includesmagnetic particles can be injected into a subject's blood stream andimages can be acquired of blood vessels that carry the magnetic particlecontrast agent.

SUMMARY

In accordance with the various embodiments of the present disclosure,there are provided devices, methods, and systems for magnetic particleimaging. Aspects of certain embodiments include a magnetic particleimaging device. The device includes a magnetic field source configuredto produce a magnetic field having a non-saturating magnetic fieldregion, an excitation signal source configured to produce an excitationsignal in the non-saturating magnetic field region that produces adetectable signal from magnetic particles in the non-saturating magneticfield region, and a signal processor configured to convert a detectedsignal into an image of the magnetic particles. Aspects of the presentdisclosure also include methods of imaging magnetic particles in asample, and methods of producing an image of magnetic particles in asubject. The subject devices and methods find use in a variety ofapplications, such as medical imaging applications.

Various embodiments of the present methods and systems will be describedin detail with reference to the drawings, wherein like referencenumerals represent like parts throughout the several views. Reference tovarious embodiments does not limit the scope of the claims attachedhereto. Additionally, any examples set forth in this specification arenot intended to be limiting and merely set forth some of the manypossible embodiments for the claims.

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art. Although any methods, devices and material similar orequivalent to those described herein can be used in practice or testing,the methods, devices and materials are now described.

All publications and patent applications in this specification areindicative of the level of ordinary skill in the art and areincorporated herein by reference in their entireties.

It should be noted that two or more of the embodiments described herein,including those described above, may be combined to produce one or moreadditional embodiments which include the combined features of theindividual embodiments. These and other aspects of the presentdisclosure will become more fully apparent from the following detaileddescription of the embodiments, the appended claims and the accompanyingfigures.

In this specification and the appended claims, the singular forms “a,”“an,” and “the” include plural reference, unless the context clearlydictates otherwise. Unless defined otherwise, all technical andscientific terms used herein have the same meaning as commonlyunderstood to one of ordinary skill in the art.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1( a) to 1(c) show schematic drawings of magnetic field sources,according to embodiments of the present disclosure.

FIG. 2( a) shows a cross-sectional illustration of a pair of magneticfield sources that produce a magnetic field gradient and anon-saturating magnetic field region in the form of a line, according toembodiments of the present disclosure. FIGS. 2( b) and 2(c) showillustrations of an MPI computed tomography technique using anon-saturating magnetic field region in the form of a line to acquireprojected image slices at different angles, according to embodiments ofthe present disclosure.

FIG. 3( a) is a graph of a received signal level versus frequency for aconventional MPI technique showing multiple received harmonics. FIG. 3(b) is a graph of received signal level versus frequency for an MPItechnique using intermodulation, according to embodiments of the presentdisclosure. FIG. 3( c) is a graph of received signal level versusfrequency detailing intermodulation tones contained in a single peak ofthe signal envelope shown in FIG. 3( c).

FIG. 4 illustrates the use of a homogeneous field superimposed on aninhomogeneous gradient field to shift the position of the field-freepoint, according to embodiments of the present disclosure.

FIG. 5 illustrates an example of a power-efficient scanning trajectory,according to embodiments of the present disclosure.

FIGS. 6( a) and 6(b) illustrate separate x, y, z receiver coils used tomeasure signals in orthogonal x, y, z directions according to anembodiment of the invention.

FIGS. 7( a) and 7(b) are schematic diagrams of the transmit and receivecircuit chains, respectively, according to embodiments of the presentdisclosure. FIG. 7( c) is a schematic diagram of the transmit circuitchain, according to embodiments of the present disclosure. FIG. 7( d) isa block diagram of a receive circuit chain, according to embodiments ofthe present disclosure.

FIGS. 8( a) and 8(b) are circuit diagrams illustrating two alternativereceive coil circuits for providing a dual-tuned receive coil, accordingto embodiments of the present disclosure.

FIGS. 9( a) and 9(b) are graphs of the transfer functions for transmitand receive circuit chains of a MPI device, according to embodiments ofthe present disclosure.

FIG. 10( a) is a cross-sectional view of an MPI apparatus, according toembodiments of the present disclosure. FIG. 10( b) is a perspectivecut-away view of a magnetic subsystem of a device, according toembodiments of the present disclosure.

FIGS. 11( a) and 11(b) are block diagrams of signal processing circuitblocks used to process the signals from the receive coil circuit chains,according to embodiments of the present disclosure.

FIG. 12 shows images showing how multiple harmonic images are alldifferent measurements of the same point source, according toembodiments of the present disclosure.

FIGS. 13( a) and 13(b) illustrate a technique of adaptivemulti-resolution scanning, according to embodiments of the presentdisclosure.

FIG. 14 shows a graph illustrating that addition of time varyinghomogeneous offset field H(t) to a gradient field −Gu causes a shiftingof the field free region x(t), according to embodiments of the presentdisclosure. For a gradient −G, the location of the FFR can be solved foras x(t)=G⁻¹H(t).

FIG. 15 shows a graph of the magnetization of the system when the FFP isat location x and SPIO nanoparticles positioned at the small circles(left), according to embodiments of the present disclosure. Themagnetization is shown for one particle and with two particles. Signalproduced by the magnetization when rapidly scanning the FFP back andforth with trajectory x(t) is shown in the graph on the right. Thesignal is shown graphed against the position of the FFP. The signalchanges in sign when the FFP is scanning in the opposite direction.

FIG. 16 shows a graph of a simulated MPI 1D image of a complex phantom(solid line) and the source distribution (dotted line), according toembodiments of the present disclosure.

FIG. 17 shows a graph of intrinsic MPI resolution for variousnanoparticle diameters when imaged using different gradient fieldstrengths, according to embodiments of the present disclosure.

FIG. 18( a) shows a graph of the relationship between bandwidth andresolution assuming a brick-wall receive filter, according toembodiments of the present disclosure. The intrinsic resolution of theMPI process may be 150% of the theoretically possible resolution whenthe receiver bandwidth is Δf_(1.5)≈2.2 F_(3dB). 110% of the intrinsicresolution is not reached until Δf_(1.1)≈3.8 F_(3dB). FIG. 18( b) showsa graph of the linear relationship between F_(3db) bandwidth for variousparticle sizes and magnetic field slew rate, according to embodiments ofthe present disclosure.

FIG. 19 shows a graph of the maximum magnetic field slew rate for a 4W/kg Specific Absorption Rate (SAR) (top), according to embodiments ofthe present disclosure. Optimal scanning frequency at the 4 W/kg RGlimit is shown in the bottom graph.

FIG. 20 shows a schematic cross-section of a MPI spectrometer fortesting point spread function and bandwidth of the MPI process,according to embodiments of the present disclosure. The excitationmagnet generated a 160 mT peak-to-peak oscillating magnetic field at6.23 kHz. The bias coil supplied a DC magnetic field of up to ±80 mT.The signal received from the gradiometric receive coil was digitized at1.25 MSPS without filtering.

FIG. 21( a) shows a graph of a parametric plot of FFP position andreceived signal with multiple offset field strengths (top), according toembodiments of the present disclosure. The peaks corresponded to theoffset field generated by the bias coil. Received signal divided byinstantaneous FFP velocity is shown in the graph in FIG. 21( a)(bottom). This is equivalent to the 1D image (see Equation II.2). FIG.21( b) shows a graph of the measured PSF compared with simulated PSF ofa particle distribution, also shown with the point spread function of ahypothetical 30 nm particle (top), according to embodiments of thepresent disclosure. Log-normal particle size distribution used togenerate simulated data is shown in the graph in FIG. 21( b) (bottom).

FIG. 22 shows a schematic drawing of two opposing ring magnets withradial symmetry about the z-axis produce a 3D gradient field with aField Free Point (FFP) at the isometric center, according to embodimentsof the present disclosure. The imaging device produced a gradient of 6T/m in the z-axis, and 3 T/m in the x- and y-axes across a 8.89 cm freebore through the z-axis with linearity.

FIG. 23 shows graphs of the Tangential and Normal Point Spread Functionenvelopes, ENV_(T) and ENV_(N) shown for ∥kH∥≦20, according toembodiments of the present disclosure. ENV_(T) is the limit to MPIresolution, and defines MPI bandwidth. ENV_(N) has approximately halfthe intrinsic resolution with FWHM_(T)=4.2 and FWHM_(N)=9.5. The valuekH is unitless.

FIG. 24 shows graphs of collinear and transverse components of thematrix point spread function, according to embodiments of the presentdisclosure. The received images rotate with vector

(see FIG. 25). The collinear PSF component peak amplitude is 370% thetangential PSF component peak amplitude. The area of the box drawn inthe collinear PSF is experimentally measured, as shown in FIG. 28.

FIG. 25 shows a graph representing that MPI images are acquired on areference frame formed by vectors collinear and transverse to thevelocity vector

, according to embodiments of the present disclosure.

FIG. 26( a) shows a schematic of a tomographic MPI scanner with 2 cm×2cm×4 cm Field of View, according to embodiments of the presentdisclosure. The excitation transmit coil generated a 30 mT peak-to-peakoscillating magnetic field at 20 kHz. The NdFeB magnet gradientgenerated a gradient of 6 T/m down the imaging bore, and 3.25 T/mtransverse to the imaging bore. FIG. 26( b) and FIG. 34 show photographsof an x-space MPI scanner, according to embodiments of the presentdisclosure. The free bore before addition of the transmit and receivecoils was 8.4 cm.

FIG. 27( a) shows a graph of experimental data showing 40 overlappingpartial field of view line-scans for a 400 micron point source phantom,according to embodiments of the present disclosure. The baselinecomponent for each partial FOV was lost in the scanning process due tothe contamination of first harmonic imaging data by direct feedthrough.FIG. 27( b) shows a graph using standard image processing methods toreconstruct a smooth version of the data segments, obtaining themaximally continuous image, according to embodiments of the presentdisclosure.

FIG. 28( a) shows a graph of measured two-dimensional collinear PSFshowing correspondence to FIG. 24, according to embodiments of thepresent disclosure. The measured FWHM was 1.6 mm along the imager boreand 7.4 mm transverse to the imager bore. The PSF phantom was a 400micron tubing oriented perpendicular to the bore. FIG. 28( b) shows agraph of theoretical PSF assuming SPIO nanoparticle of lognormal sizedistribution with d=17±3.4 nm, according to embodiments of the presentdisclosure.

FIGS. 29( a) and 29(b) shows graphs of profiles across the point spreadfunction shown in FIGS. 28( a) and 28(b), which show good agreementbetween theoretical and measured values, according to embodiments of thepresent disclosure. FIG. 29( a) shows a graph of a line scan down thebore, and FIG. 29( b) shows a graph of a line scan perpendicular to theimager bore, according to embodiments of the present disclosure.

FIG. 30( a) shows a photograph of a “CAL” phantom image built using 400micron ID tubing filled with undiluted tracer and encapsulated,according to embodiments of the present disclosure. FIG. 30( b) shows anintrinsic MPI image of the CAL phantom showing correspondence to thephantom image, according to embodiments of the present disclosure. TheFOV was 4 cm×2 cm, and the pixel size was 200 micron×1 mm. The totalimaging time was 28 seconds, not including robot movement.

FIG. 31 shows a schematic of a tomographic MPI scanner with 2 cm×2 cm×4cm Field of View constructed to test the x-space formulation for MPI,according to embodiments of the present disclosure. The excitation coilgenerated a 30 mT peak-to-peak oscillating magnetic field at 19 kHz. TheNdFeB magnet gradient generated a gradient of G_(z)=6.0 T/m down theimaging bore, and G_(x,y)=3.0 T/m transverse to the imaging bore. Thesample was mechanically moved through the bore.

FIG. 32( a) shows a graph of a theoretical collinear PSF component for a4 cm×4 cm FOV with 6 T/m×3 T/m gradient, according to embodiments of thepresent disclosure. The high gradient was collinear with the FFPvelocity vector

. FIG. 32( b) shows a graph of Transverse and Normal PSF envelopes forthe same system, according to embodiments of the present disclosure. TheNormal envelope had a FWHM that was 460% wider.

FIG. 33( a) shows a graph of an x-space pulse sequence, according toembodiments of the present disclosure. FIG. 33( b) shows a graph of apulse sequence used in the x-space scanner with overlap of 50%,according to embodiments of the present disclosure. Rapid movement in zwas performed electronically at 20 kHz, while slow movement in x, y andz was performed mechanically. In certain embodiments, the x-spacescanner may slowly move the FFP electronically.

FIG. 34 shows a photograph of x-Space MPI scanner, according toembodiments of the present disclosure. The free bore before addition ofthe transmit and receive coils was 8.4 cm.

FIG. 35 shows a schematic of the Transmit-Receive electronics, accordingto embodiments of the present disclosure.

FIGS. 36( a) and 36(b) show graphs of measured signal showing phasecorrected signal from a single scan across a point source in z and y,according to embodiments of the present disclosure. FIG. 36( a) is agraph showing that the amplitude changed slowly as the sample wasscanned 1.5 cm in y. FIG. 36( b) shows a graph of a time-slice near y=0showing the raw signal as the sample was rapidly scanned 0.5 cm in z.Total scan time was 650 ms.

FIG. 37( a) shows a graph of experimental data showing 40 overlappedpartial FOV scans of a 400 micron wide Resovist point source phantomwithout baseline correction, according to embodiments of the presentdisclosure. FIG. 37( b) shows a graph of experimental data with baselinecorrection, and FIG. 37( c) shows a graph of the assembled image thatrecovers the linearity across the full FOV, according to embodiments ofthe present disclosure.

FIGS. 38( a) and 38(b) show graphs of a comparison between measured andtheoretical collinear component of the PSF, according to embodiments ofthe present disclosure. The measured FWHM was 1.6 mm along the imagerbore and 7.4 mm transverse to the imager bore. The field of view was 4cm×2 cm, and the total imaging time was 28 seconds, not including robotmovement.

FIGS. 39( a) and 39(b) show graphs of profiles across the point spreadfunction, which show good agreement between theoretical and measuredvalues, according to embodiments of the present disclosure. TheoreticalPSF assuming SPIO nanoparticle of lognormal size distribution withd=17±4 nm. The PSF phantom was a 400 micron ID tubing filled withResovist oriented perpendicular to the bore. FIG. 39( a) shows a graphof a line scan down the bore, and FIG. 39( b) shows a graph of a linescan perpendicular to the imager bore, according to embodiments of thepresent disclosure.

FIG. 40 shows a graph of a line scan of a linear Resolution phantom withpoint sources separated by 1 mm, 2 mm, and 3 mm, according toembodiments of the present disclosure. The 1 mm spaced samples were notresolvable as the spacing between them was less than the intrinsicresolution of the system (FWHM=1.6 mm).

FIG. 41 shows a graph of a measured power spectrum of signal compared totheoretical power spectrum of signal assuming excitation amplitude of30/√{square root over (2)} mT_(p-p), according to embodiments of thepresent disclosure. The harmonics were an artifact of repeatedlyscanning over the sample. The measured signal corresponds to theory.

FIG. 42 shows a graph of an image of phantom (solid line) composed ofsample composed of linearly increasing quantities of iron oxide tracer(1, 2, and 3) superimposed with the integral of the line scan (dottedline), according to embodiments of the present disclosure. The phantomdemonstrates that signal was linear with the quantity of iron oxidetracer.

FIG. 43( a) shows a photograph of a “CAL” phantom image built using 400micron ID tubing filled with undiluted tracer and encapsulated,according to embodiments of the present disclosure. FIG. 43( b) showsthe intrinsic MPI image of the CAL phantom showing correspondence to thephantom image, according to embodiments of the present disclosure. FIG.43( c) shows a Wiener filtered image of intrinsic image, according toembodiments of the present disclosure. The FOV was 4 cm×2 cm, and thepixel size was 200 micron×1 mm. The total imaging time was 28 seconds,not including robot movement.

FIG. 44( a) shows a photograph of a preserved chicken phantom, and FIG.44( b) shows a Wiener filtered image of the phantom image, according toembodiments of the present disclosure. The chicken tissue itself doesnot appear in the MPI image, thus there is no background to the image.

FIG. 45 shows a graph of a raw signal that has a smooth power spectrumdensity, according to embodiments of the present disclosure.

FIG. 46 shows a graph of repeating scanning of the sample introducesharmonics, according to embodiments of the present disclosure.

FIG. 47 shows a photographic image of the small scale MPI scannershowing three axis robot on left, and the scanner bore in the center,according to embodiments of the present disclosure.

FIG. 48( a) shows an image of measured signal power (in dB) as afunction of position of a point source, and FIG. 48( b) shows a graph ofmaximum measured signal power over all point source positions, accordingto embodiments of the present disclosure. FIG. 48( c) shows a graph ofthe log of the measured signal power, according to embodiments of thepresent disclosure. The theoretical MPI signal bandwidth was comparableto the measured signal when assuming RMS FFP movement speed. The linearscale was logical for viewing the signal, as continuing to increase thereceive bandwidth beyond 200 kHz did not improve the resolution. Thetheoretical particle in the bandwidth calculation had a lognormaldistribution with mean diameter and standard deviation d=18±1.5 nm. Thesecond harmonic was slightly reduced due to analog filtering of thefundamental frequency.

FIG. 49( a) shows a photographic image of a resolution phantom composedof polyethylene tubing (ID=400 μm) filled with undiluted Resovisttracer, according to embodiments of the present disclosure, and FIG. 49(b) shows graphs of baseline corrected, unassembled data for a widebandwidth and a narrow bandwidth. FIG. 49( c) shows a graph of theassembled data showing the resolution phantom of the reconstructed imagewith three different low pass filters on the raw data before gridding,according to embodiments of the present disclosure. The image was takenwith a 5 mm partial FOV with 80% overlap. Total scan time not includingrobot movement was 0.8 seconds.

FIG. 50 shows a graph of measured FWHM as a function of bandwidth,according to embodiments of the present disclosure. The point source wasundiluted Resovist in polyethylene tubing with ID=400 μm. Increasing thebandwidth above 200 kHz did not substantially increase resolution.

FIG. 51 shows a graph of the measured RMS noise of a scan post-griddingimproves with decreasing system bandwidth, according to embodiments ofthe present disclosure.

FIG. 52 shows a photographic image of a magnetic particle imaging deviceconfigured to produce a non-saturating magnetic field line and includetwo permanent magnets positioned on opposing sides of the centralimaging area of the device, according to embodiments of the presentdisclosure. The device generated a 2.5 T/m gradient across a 4 inchmagnet free bore.

FIG. 53 shows a schematic of the position of two permanent magnets in amagnetic particle imaging device configured to produce a non-saturatingmagnetic field line, according to embodiments of the present disclosure.

FIG. 54( a) shows a photograph of a “UC” phantom image built usingtubing filled with undiluted tracer and encapsulated, according toembodiments of the present disclosure. FIG. 54( b) shows an intrinsicMPI image of the “UC” phantom image showing correspondence to thephantom image, where the phantom image was obtained using a magneticparticle imaging device configured to produce a non-saturating magneticfield line, according to embodiments of the present disclosure. Thetotal imaging time was 12 seconds, not including robot movement.

The publications discussed herein are provided solely for theirdisclosure prior to the filing date of the present application. Nothingherein is to be construed as an admission that the present invention isnot entitled to antedate such publication by virtue of prior invention.Further, the dates of publication provided may be different from theactual publication dates which may need to be independently confirmed.All publications mentioned herein are incorporated herein by referenceto disclose and describe the methods and/or materials in connection withwhich the publications are cited.

DETAILED DESCRIPTION Magnetic Particle Imaging Devices

As described in further detail below, in accordance with the variousembodiments of the present disclosure, there are provided devices,methods, and systems for magnetic particle imaging (MPI). Aspects ofcertain embodiments include a magnetic particle imaging device. Themagnetic particle imaging device may produce an image of magneticparticles in a sample.

Magnetic Field Source

The magnetic particle imaging device may include a magnetic fieldsource. The magnetic field source may be of sufficient strength tosaturate magnetic particles present in the magnetic field produced bythe magnetic field source. Stated another way, the magnetic field sourcemay produce a saturating magnetic field, where the saturating magneticfield has a sufficient magnetic field strength to saturate magneticparticles present in the saturating magnetic field. By “saturate” or“saturation” is meant that the magnetic particles have a magnetizationsuch that an increase in the applied external magnetizing field will notsignificantly increase the magnetization of the magnetic particlesfurther. In some instances, a magnetic particle includes a plurality ofmagnetic domains, each with a corresponding magnetic field. Applicationof an external magnetic field source to the magnetic particles may causethe magnetic fields of the magnetic domains to align parallel to theapplied external magnetic field. In some instances, the applied externalmagnetic field is of sufficient strength to “saturate” the magneticparticle, such that substantially all of the magnetic domains in themagnetic particle are aligned parallel to the applied external magneticfield, so further increases in the applied external magnetic field willnot substantially cause further alignment of the magnetic domains.

In certain embodiments, the magnetic field source is configured toproduce a magnetic field having a non-saturating magnetic field region.The non-saturating magnetic field region may be positioned within aportion of the magnetic field. By “non-saturating” is meant that theapplied magnetic field in that region has a strength below thatnecessary to saturate the magnetic particles in the non-saturatingmagnetic field region. In certain embodiments, the non-saturatingmagnetic field region has a magnetic field strength of substantiallyzero. The non-saturating magnetic field region may also be referred toas a “field-free region” or “FFR”. The non-saturating magnetic fieldregion may be configured as a point, a line, a plane, or a 3-dimensionalregion. In some instances, the non-saturating magnetic field region is aline. In these cases, the non-saturating magnetic field line may beperpendicular to the axis of the magnetic field. In some instances, thenon-saturating magnetic field line is substantially parallel to the axisof the magnetic field. In other embodiments, the non-saturating magneticfield region is a 3-dimensional region of space in the magnetic field.

In certain embodiments, the magnetic field source includes two or moremagnetic field sources, such as 4 or more, or 6 or more, or 8 or more,or 10 or more magnetic field sources. The two or more magnetic fieldsources may be configured such that the two or more magnetic fieldsources have a combined magnetic field. In certain cases, the combinedmagnetic field is a saturating magnetic field. In some instances, thecombined magnetic field includes a non-saturating magnetic field region,(e.g., a field-free region).

The magnetic field sources may be arranged in various orientationsrelative to each other. For example, the magnetic field sources may bearranged relative to each other such that the combined magnetic fieldproduced by the magnets includes a non-saturating region in the magneticfield. The magnetic field sources may be square shaped, rectangular,circular, elliptical, spherical, ring-shaped, combinations thereof, andthe like. In certain embodiments, the magnetic field sources arering-shaped. In these embodiments, the magnetic field sources may becoaxially arranged, such that the center of each ring-shaped magneticfield source is on the same axis (e.g., the coaxial axis). In someinstances, the coaxial axis passes through the center of eachring-shaped magnetic field source and is substantially perpendicular toeach magnetic field source. In certain embodiments, the coaxial axis islabeled as the z-axis in a 3-dimensional coordinate system, and isperpendicular to the x and y axes, such that the x-axis is perpendicularto both the y and z axes, the y-axis is perpendicular to both the x andz axes, and the z-axis is perpendicular to both the x and y axes.

In some cases, the magnetic field sources are arranged around an imagingarea of the device. The imaging area of the device may be configured tocontain a sample that is to be imaged and, in some cases, is positionedbetween the magnetic field sources. For example, in embodiments wherethe magnetic field sources are ring-shaped magnetic field sources asdescribed above, the coaxial axis of the magnetic field sources may besubstantially parallel to a longitudinal axis of an imaging area of thedevice. The magnetic field sources may be positioned at opposing ends ofthe imaging area of the device. For instance, a first magnetic fieldsource may be positioned at one end of the longitudinal axis of theimaging area of the device, and a second magnetic field sourcepositioned at the opposite end of the longitudinal axis of the imagingarea of the device.

In other embodiments, the magnetic field sources are arranged on thesides of the imaging area of the device (rather than at each end of theimaging area of the device). For instance, a first magnetic field sourcemay be positioned on one side of the imaging area of the device, such asalong a side of the imaging area substantially parallel to alongitudinal axis of the imaging area of the device. A second magneticfield source may be positioned on an opposing side from the firstmagnetic field source, such as along a side of the imaging area of thedevice opposite the first magnetic field source and substantiallyparallel to the longitudinal axis of the imaging area of the device.

The magnetic field source may include permanent magnets, electromagnets,superconducting magnets, high-mu materials (e.g., iron), combinationsthereof, and the like. In certain embodiments, the magnetic field sourceincludes one or more permanent magnets. By “permanent magnet” is meant amagnetic material has a persistent magnetic field such that the magneticfield that does not substantially decrease over time. In contrast, theterm “soft magnet” refers to a material that can be magnetized in thepresence of an applied external magnetic field, but whose magnetismsubstantially decreases when the external magnetic field is removed. Insome instances, the magnetic field source includes two or more permanentmagnets. The permanent magnets may be of any desirable shape, and insome instances may be ring-shaped permanent magnets as described above.The ring-shaped permanent magnets may be coaxially arranged relative toeach other.

The magnetic field source may be a permanent magnet, such as arare-earth magnet. Rare-earth magnets include, but are not limited to,samarium-cobalt magnets (e.g., SmCO₅), neodymium alloy (NdFeB) magnets(e.g., Nd₂Fe₁₄B), and the like.

In certain embodiments, the magnetic field source produces a magneticfield ranging from 0.01 mT to 25 T, or from 0.01 mT to 10 T, or from0.01 mT to 5 T, or from 0.01 mT to 3 T, or from 0.01 mT to 1 T, such asfrom 0.1 mT to 500 mT, including from 1 mT to 100 mT, for example, from1 mT to 30 mT, or from 10 mT to 20 mT. In certain cases, the magneticfield sources produce an inhomogeneous magnetic field. By“inhomogeneous” is meant that the magnetic field is different dependingon the position within the magnetic field. For instance, the magneticfield may have a magnetic field gradient that is greater at one positionin the magnetic field and gradually decreases towards a second positionin the magnetic field. In some cases, the magnetic field sources areconfigured to produce a magnetic field with a magnetic field gradientranging from 0.1 T/m to 250 T/m, such as from 0.1 T/m to 100 T/m, orfrom 0.1 T/m to 75 T/m, or from 0.1 T/m to 50 T/m, such as from 0.5 T/mto 40 T/m, including from 0.5 T/m to 30 T/m, or from 1 T/m to 30 T/m,for example from 1 T/m to 20 T/m, or from 1 T/m to 10 T/m, or from 1 T/mto 7 T/m, or from 2.5 T/m to 7 T/m. In certain instances, the magneticfield sources produce a magnetic field with the same magnetic fieldgradient along the coaxial axis as along an axis transverse to thecoaxial axis. In some cases, the magnetic field sources produce amagnetic field with a different magnetic field gradient along thecoaxial axis from the magnetic field gradient along an axis transverseto the coaxial axis. For example, the magnetic field gradient along thecoaxial axis may be 1.2 times greater, or 1.4 times greater, or 1.6times greater, or 1.8 times greater, or 2 times greater, or 3 timesgreater, or 4 times greater, or 5 times greater than the magnetic fieldgradient along an axis transverse to the coaxial axis.

Various designs and configurations of magnetic field sources may be usedin various embodiments. Examples of magnetic field sources areillustrated in FIGS. 1( a), 1(b), 1(c). A front entry design using ringmagnets 100, 102 is shown in FIG. 1( a). Also shown are cut-away viewswhere the magnetic field sources are permanent magnets 104, orelectromagnets 106. A side entry design using circular plate magnets110, 112 is shown in FIG. 1( b). Also shown are cut-away views where themagnetic field sources are disc-shaped permanent magnets 114, orelectromagnets 116. FIG. 1( c) shows a front entry design using a pairof Halbach arrays 120, 122.

FIG. 2( a) is a cross-sectional illustration of a pair of magnetic fieldsources 200 and 202 that produce a strong field gradient and afield-free region 204. The magnetic field sources can be a permanentmagnet arrangement having a three dimensional structure that can bemachined from a single block of permanent magnet. In some cases, thepermanent magnets are designed using an L¹-norm optimization method. Insome embodiments, the magnetic field sources are designed so that thereis an axial entry and the field free region has a longitudinal axisperpendicular to the axis of the magnetic field. The magnet or thesample being imaged can be rotated mechanically.

Excitation Signal Source

Aspects of embodiments of the magnetic particle imaging device includean excitation signal source configured to produce an excitation signalin the non-saturating magnetic field region. In some cases, theexcitation signal source is configured to produce an excitation signalin the non-saturating magnetic field region sufficient to produce adetectable signal from magnetic particles in the non-saturating magneticfield region. For example, the excitation signal source may beconfigured to apply the excitation signal to magnetic particles in thenon-saturating magnetic field region. In some instances, application ofthe excitation signal to the magnetic particle in the non-saturatingmagnetic field region produces a detectable signal from the magneticparticles in the non-saturating magnetic field region.

The excitation signal source may include a radio frequency (RF)excitation signal source that produces an RF excitation signal. The RFexcitation signal source may produce a magnetic field, which in someinstances is an oscillating magnetic field. The RF excitation signalsource may be configured to produce a magnetic field ranging from 0.1 mTto 5 T peak to peak, or from 0.1 mT to 3 T peak to peak, or from 0.1 mTto 1 T peak to peak, or from 0.1 mT to 500 mT peak to peak, or from 0.1mT to 250 mT peak to peak, or from 1 mT to 100 mT peak to peak, or from1 mT to 50 mT peak to peak, such as from 10 mT to 50 mT peak to peak,including from 20 mT to 40 mT peak to peak. In some cases, the RFexcitation signal source is configured to produce a magnetic field of 30mT peak to peak. In certain embodiments, the RF excitation signal sourceproduces an oscillating magnetic field having a frequency ranging from0.1 Hz to 1000 MHz, or from 1 Hz to 500 MHz, or from 1 kHz to 250 MHz,or from 1 kHz to 100 MHz, or from 1 kHz to 50 MHz, or from 1 kHz to 25MHz, or from 1 kHz to 10 MHz, such as from 10 kHz to 10 MHz, includingfrom 10 kHz to 1 MHz, for example from 10 kHz to 500 kHz, or from 10 kHzto 100 kHz. In some instances, the RF excitation signal source producesan oscillating magnetic field having a frequency of 20 kHz.

The RF excitation signal may be a periodic oscillating field, such as asinusoidal waveform. However, in some instances, the waveform of the RFexcitation signal is not sinusoidal. In some cases, a non-sinusoidalwaveform may facilitate an increase in harmonic content, improving thesignal to noise ratio (SNR) and resolution. For example, certainembodiments of the RF excitation signal may include a triangle waveform.The waveform may also be dynamically changed during operation to providedifferent imaging properties.

The excitation signal source may include an intermodulation excitationsignal source that produces an intermodulation excitation signal. Theintermodulation excitation signal source may produce a magnetic field,which in some instances is an oscillating magnetic field. Theintermodulation excitation signal source may be configured to produce amagnetic field ranging from 0.1 mT to 1 T peak to peak, or from 0.1 mTto 500 mT peak to peak, or from 0.1 mT to 250 mT peak to peak, or from 1mT to 100 mT peak to peak, or from 1 mT to 75 mT peak to peak, 1 mT to50 mT peak to peak, such as from 1 mT to 40 mT peak to peak, includingfrom 1 mT to 30 mT peak to peak, or from 1 mT to 20 mT peak to peak, orfrom 1 mT to 10 mT peak to peak. In some cases, the intermodulationexcitation signal source is configured to produce a magnetic field of 6mT peak to peak. In certain embodiments, the intermodulation excitationsignal source is a low frequency (LF) intermodulation excitation signalsource that produces an oscillating magnetic field having a frequencyranging from 1 Hz to 1 MHz, or from 1 Hz to 500 kHz, or from 1 Hz to 250kHz, or from 1 Hz to 100 kHz, or from 1 Hz to 50 kHz, or from 1 Hz to 20kHz, such as from 1 Hz to 10 kHz, including from 1 Hz to 5 kHz, forexample from 1 Hz to 1 kHz, or from 1 Hz to 500 Hz. In some instances,the intermodulation excitation signal source produces an oscillatingmagnetic field having a frequency of 1 kHz. In certain embodiments, theRF excitation signal has a frequency that is greater than the frequencyof the intermodulation excitation signal. For example, the RF excitationsignal may have a frequency that is 5 to 1,000,000 times greater thanthe frequency of the intermodulation excitation signal, such as 50 to100,000 times greater, including 100 to 10,000 times greater, or 100 to5,000 times greater, or 100 to 500 times greater than the frequency ofthe intermodulation excitation signal.

Intermodulation Theory

When magnetic field strengths used in MPI are less that 1 Tesla, tissueis unaffected by the magnetic field, but a super-paramagnetic iron oxide(SPIO) particle undergoes a nonlinear change in magnetization describedby the Langevin theory of paramagnetism. Specifically, the magnetizationM is given by:

$M = {{m_{0}{L\left\lbrack \frac{m\; H}{k_{n}T} \right\rbrack}} = {M_{0}\left( {{\coth \frac{m\; H}{k_{B}T}} - \frac{k_{B}T}{m\; H}} \right)}}$

where L is the Langevin function, m is the magnetic moment of theparticle, H is the applied magnetic field, k_(B) is Boltzmann'sconstant, and T is the absolute temperature.

To excite the particles, in some embodiments a single oscillatingmagnetic field of magnitude H₀ and frequency f₀ is generated within theregion where the particles are located. In the case of a sinusoidalexcitation waveform, the oscillating field is given by

H(t)=H ₀ sin(2πf ₀ t).

The field H(t) excites the particles and induces a correspondingtime-varying magnetization at harmonics of f₀

${{M(t)} = {\sum\limits_{m \geq 1}{A_{m}{\exp \left( {2\pi \; i\; m\; f_{0}t} \right)}}}},$

where A_(m) are the amplitudes of the various harmonics and the index mranges over the detected harmonics. See FIG. 3( a).

In some embodiments using intermodulation, a second oscillating magneticfield of magnitude H₁ and frequency f₁ is also generated within theregion where the particles are located. Thus, in the case of asinusoidal excitation waveform aligned in parallel with the firstexcitation field, the net oscillating field is given by

H(t)=H ₀ sin(2πf ₀ t)+H ₁ sin(2πf ₁ t).

This intermodulation field H(t) excites the particles the nonlinearLangevin function acts as a nonlinear mixer, inducing a correspondingtime-varying magnetization

${M(t)} = {\sum\limits_{m \geq 1}{\sum\limits_{n}{A_{m,n}{{\exp \left( {2{{\pi }\left( {{m\; f_{0}} + {n\; f_{1}}} \right)}t} \right)}.}}}}$

where A_(m,n) are the amplitudes of the separate intermodulation tones.In addition to the harmonics, there are sideband tones corresponding tosum and difference frequencies. See FIGS. 3( b) and 3(c). The index mmay be limited to a finite number of detected harmonics and the index nmay be limited by the finite number of detected sideband intermodulationtones around each harmonic. The intermodulation spectrum shows thequantity of magnetic particles at the field-free point. Thus, instead ofdetecting a sequence of harmonics across a broad bandwidth, it ispossible to obtain sufficient information by detecting theintermodulation signals across a relatively narrow bandwidth in closeproximity to a single harmonic. Detecting intermodulation signals aroundadditional harmonics provides additional information. For example,intermodulation sidebands may be detected around both second and thirdharmonics to produce a set of intermodulation images.

The frequencies f₀ and f₁ and the field strengths H₀ and H₁ can all beselected independently of each other. In some cases, the specificabsorption rate (SAR) and received signal strength depend on f₀ and H₀since f₀>>f₁ and SAR increases as H²f². Imaging speed and detectionbandwidth may depend on f₁ to allow detection at each field-free pointof intermodulation sidebands surrounding harmonics mf₀. Thus, thescanning speed may be selected so that the sidebands can be detected ateach point without aliasing. In some cases, increasing f₁ may facilitatean increase in imaging speed. In certain instances, f₁ is not increasedto a value such that the received signal bandwidth is greater than thebandwidth of the receiver coil. SNR and the spatial extent of the pointspread function (PSF) may depend on the total magnitude of theexcitation fields H_(tot)=H₀+H₁. In some cases, increasing H_(tot)increases the total signal received while also widening the PSF.Increasing H₁ may increase the received signal while not substantiallyaffecting SAR. In some instances, increasing H₁ facilitates an increasein signal while decreasing resolution, and vice versa.

LF Intermodulation

In various embodiments, the LF (low frequency) intermodulation sourcemay include one or more electromagnets, such as water-cooledelectromagnets. MRI gradient amplifiers may be used to drive themagnets. In certain embodiments, the LF intermodulation provides a largeshift in magnetization of the magnetic particles in the non-saturatingmagnetic field region while not substantially increasing the specificabsorption rate (SAR). LF intermodulation may also facilitate applyingmagnetic energy to the sample which is then up-mixed with the RFfrequency. An RF eddy current shield may be used prevent interactionbetween the LF circuits and the RF transmit coil. In addition, a set offilters, such as common mode and differential low-pass filters, may alsobe positioned between the power amplifier and coil to reduceinteraction. The gradient amplifiers may be current controlled and eddycurrent compensated.

The intermodulation excitation field may be applied in the x, y, zdirections or any subset thereof. Changing the direction theintermodulation excitation source may change the shape and magnitude ofthe point spread function.

In certain instances, the LF intermodulation excitation signal and thescanning magnetic field (described in more detail below) may both begenerated by the same magnetic field source (e.g., electromagnets). Inother embodiments, separate magnetic field sources (e.g.,electromagnets) are used to generate the LF intermodulation excitationsignal and the scanning magnetic field. Having a separate LFintermodulation excitation signal source and a separate scanningmagnetic field source may facilitate applying an LF intermodulationexcitation signal that has a different magnetic field strength and/ormagnetic field gradient from the scanning magnetic field.

Scanning Magnetic Field Source

Aspects of the magnetic particle imaging device include a scanningmagnetic field source configured to produce a scanning magnetic field.In some cases, the scanning magnetic field is configured to position thenon-saturating magnetic field region in the magnetic field. The scanningmagnetic field may be applied to the magnetic field causing thenon-saturating magnetic field region of the magnetic field to bedisplaced from its initial position in the magnetic field.Alternatively, or in addition to the scanning magnetic field source, thenon-saturating magnetic field region may be scanned through the sampleby moving the sample relative to the magnetic field. For example, thedevice may include a sample holder that may be displaced in one or moredirections relative to the magnetic field sources.

The scanning magnetic field source may be configured to cause thenon-saturating magnetic field region to scan through the sample in thedevice in one or more directions. Movement of the non-saturatingmagnetic field region relative to the sample may be performed in one,two, or three dimensions, and may be implemented by mechanical movementof the sample relative to the magnetic field source and/or byelectronically modifying the magnetic field (e.g., the inhomogeneousgradient field) using a scanning magnetic field source (e.g., scanningelectromagnets) that generate a homogeneous field. The scanning magneticfield can be applied in x, y, and z directions, or a subset of thesedirections. For example, the scanning magnetic field source (e.g.,scanning electromagnets) may be configured to position thenon-saturating magnetic field region within a plane, while mechanicalmovement of the sample provides translation along an axial directionperpendicular to the plane.

The scanning magnetic field may be a homogeneous field. In some cases,the scanning magnetic field source includes one or more electromagnets,such as but not limited to Helmholtz coils. The homogeneous field 400superimposed on the inhomogeneous gradient field 402 has the effect ofshifting the position of the non-saturating magnetic field region 404 bya distance D, as illustrated in FIG. 4. A switching amplifier such as anH-bridge may be used to reverse the direction of the homogeneousscanning magnetic field to provide displacement of the non-saturatingmagnetic field region in opposite directions along a given axis. In someembodiments, the LF intermodulation excitation source is the same as thescanning magnetic field source. In other embodiments, the LFintermodulation excitation source is different from the scanningmagnetic field source.

Due to the large field gradients, strong homogeneous fields may berequired to shift the field-free point a distance to provide electronicscanning of the sample. Consequently, scanning a large field of view canrequire large amounts of power and large amplifiers. Accordingly,certain embodiments provide methods of scanning that reduce heat loadingof the power amplifiers, allowing the use of smaller and less expensivepower electronics. For example, FIG. 5 illustrates an example of apower-efficient scanning trajectory in the x-y plane according toembodiments of the present disclosure. The scan begins with aleft-to-right scan line 500 along the x-axis. Line 500 is displaced arelatively large distance in the y-direction from center line 516.Center line 516 is the line along which the field-free region ispositioned when there is no displacement field in the y-direction.Horizontal scan line 500 is followed by a diagonal scan line down and tothe left to the start of horizontal scan line 502 which is displaced arelatively small distance in the y-direction from center line 516, andthus requires a less powerful field than is required to displace scanline 500 from center line 516. Horizontal scan line 502 is followed by adiagonal scan up and to the left to the start of horizontal scan line504, which is displaced slightly less from center line 516 than scanline 500. Next is scan line 506, which is displaced a slightly greaterdistance from the center line 516 than scan line 502. The scan thencontinues in this way, alternating between scan lines above and belowcenter line 516. The consecutive scan lines above and below the centerline 516 are separated by approximately the same distance due to theprogressive decrease in displacement of y-displacement of scan linesabove the center line 516 and progressive increase of y-displacement ofscan lines below the center line 516. Consequently, as power demands forscan lines displaced in one direction decrease, power demands in scanlines displaced in the other direction increase, resulting in anapproximately constant average power demand. This constant average powercorrelates to the uniform distance between consecutive scan line pairs,i.e., the displacement between lines 500 and 502 is the same as thedisplacement between lines 504 and 506, between lines 508 and 510, andbetween lines 512 and 514. Switching between displacements above andbelow the center line 516 may allow circuits used for displacements inone direction to cool during a scan displaced in the opposite direction.Depending on the application and operational parameters of a given scan,the scan time for each line may range from 0.5 ms to 10 sec, such asfrom 1 ms to 5 s, including from 1 ms to 1 s, for example from 1 ms to0.5 s. Because the horizontal scan lines all scan in the same direction,e.g., from left to right, this may facilitate simpler signal processing.

Other scanning trajectories are also possible and may be used dependingon the specific application or scanning requirements. For example, toreduce transition time from scan line to scan line, instead of moving tothe next scan line diagonally, in some cases only the y-displacement ischanged when moving to the next scan line. The amplifiers may have slewrate limits, and only changing the y-displacement when moving from onescan line to the next scan line may reduce the slew rate requirements inthe x-direction. For example, if heating is not a significant issue thena more time-efficient scan may be used, such as a serpentine scan ofhorizontal lines sequentially progressing from a largest upwarddisplacement in the y-direction to a largest downward displacement inthe y-direction. In some instances, a spiral scan may be performed,starting from the center and spiraling outward. The spiral scan mayfacilitate scanning a sample region with a greater amount of overlapbetween scans than a rectangular scanning pattern.

In certain embodiments, the scanning magnetic field source translatesthe position of the non-saturating magnetic field region in real time toprovide scanning. The magnetic particle imaging device may be configuredto provide for data acquisition and magnetic particle detection in realtime as the scanning magnetic field positions the non-saturatingmagnetic field region through the sample. In some embodiments, a pair ofelectromagnet coils is used to provide independent translation in eachof three orthogonal directions. The scanning magnetic field source mayhave a power of 0.1 kW to 1 MW, or from 0.1 kW to 500 kW, or from 0.1 kWto 250 kW, or from 0.1 kW to 100 kW, such as 1 kW to 50 kW, including 1kW to 10 kW. In some instances, the scanning magnetic field source has apower of 5 kW. In certain cases, the scanning magnetic field sources areconfigured to move the non-saturating magnetic field region from 0.1 cmto 100 cm, such as from 0.1 cm to 50 cm, including from 0.1 cm to 25 cm,or from 0.1 cm to 10 cm, or from 0.1 cm to 5 cm from the initialposition of the non-saturating magnetic field region. In some instances,the scanning magnetic field sources are configured to produce a scanningmagnetic field with a homogeneity of 10% or less, such as 5% or less,including 3% or less, for example 1% or less.

Other systems associated with the excitation signal sources may beincluded in the device. For example, to provide cooling, the excitationsignal sources may be configured as electromagnetic coils. In someembodiments, the electromagnetic coils are configured to be cooled bycontacting the coils with a coolant. For instance, the coils may be madeof hollow copper tubing through which coolant (e.g., water) may becirculated. In some instances, the coolant may be circulated at a rateto provide sufficient cooling capacity for the excitation coils. Forinstance, the coolant (e.g., water) may be circulated at 6 gpm and 30psi, which in some instances provides 34 kW cooling capacity. Otheraspects of the excitation signal source may include a shield, such as anRF shield. The RF shield may be configured to absorb signals that arenot involved in the excitation of the magnetic particles and/or thesignal generated by the magnetic particles. In some instances, the RFshield is configured to be cooled using circulating coolant (e.g.,water).

Receiver

Aspects of the magnetic particle imaging device include a receiver. Thereceiver may be configured to detect the signals from the magneticparticles in the non-saturating magnetic field region. For example,after applying an excitation signal to the magnetic particle in thenon-saturating magnetic field region, as described above, the magneticparticles may produce a detectable signal, which may be detected by oneor more receivers. The receivers may be any type of receiver that iscapable of receiving the detectable signals from the magnetic particles.For example, the receiver may be configured to detect magnetic signalsfrom the magnetic particles and convert the detected magnetic signalsinto an electrical signal.

In some embodiments, the receiver includes a receiver coil. In somecases, the receiver is configured to be a narrowband receiver. Forexample, the receiver may have a quality factor (e.g., Q-factor) of 1 ormore, such as 10 or more, including 50 or more, or 100 or more, forinstance 150 or more, or 200 or more, or 500 or more, or 1000 or more.The O-factor is a dimensionless parameter that characterizes thebandwidth of the receiver relative to the center frequency of thereceiver. A receiver with a high Q-factor (e.g., 100 or more) mayfacilitate an increase in the signal to noise ratio (SNR) by reducingthe bandwidth requirements of the receiver. For instance, in some cases,the receiver is configured to have a receive bandwidth ranging from 1kHz to 100 MHz, such as from 1 kHz to 50 MHz, including from 5 kHz to 25MHz, or from 10 kHz to 10 MHz, or from 10 kHz to 5 MHz, or from 10 kHzto 1 MHz, or from 20 kHz to 500 kHz, or from 20 kHz to 200 kHz.

In certain embodiments, the receiver includes a receive coil configuredas a gradiometer. The gradiometer include a Litz wire and an overallQ-factor of 100 or more, or 150 or more (e.g., Q_(coil)=167). In somecases, the receiver coil includes an inner coil and an outer coil. Theinner coil may have a diameter less than the diameter of the outer coil.For instance, the inner coil may have a diameter ranging from 1 cm to 1m, such as from 1 cm to 500 cm, or from 1 cm to 250 cm, or from 1 cm to100 cm, or from 1 cm to 50 cm, or from 1 cm to 25 cm, or from 1 cm to 10cm. The outer coil may have a diameter greater than the diameter of theinner coil, and may range from 1 cm to 1 m, such as from 1 cm to 500 cm,or from 1 cm to 250 cm, or from 1 cm to 100 cm, or from 1 cm to 50 cm,or from 1 cm to 25 cm, or from 1 cm to 10 cm. For example, the innercoil may have a diameter of 3.175 cm and the outer coil may have adiameter of 4.5 cm.

The receiver may include additional electronics associated with thereceiver coil, such as, but not limited to, a phase coherent controlconsole and detector. The coherent detector may be configured todirectly sample at 65 MSPS and digitally down convert the RF signal tobaseband. The down-sampled signal may have a bandwidth of 31.25 kSPScentered at 2f₀=300 kHz with over 90 dB of dynamic range, where f₀ isthe excitation frequency. An object containing a distribution ofmagnetic particles may be translated through the bore using a lineartranslator controlled by the control console. The detected signal may becontinuously acquired during translation of the stage in the readoutdirection, e.g., along the axis of the bore. The signal received by thereceiver coil may be transmitted to the preamplifier and then into thecontrol console. The digitized signal may be quadrature demodulated atmultiples of the intermodulation frequency (i.e., ±f₁, ±2f₁, ±3f₁, ±4f₁,etc.) and brick-wall filtered at 20 Hz, where f₁ is the intermodulationfrequency. In some instances, the intermodulation products around thefundamental frequency may be more difficult to receive than those aroundthe harmonics due to eddy-current coupling from the excitation frequencyinto the receive coil. The intermodulation products around thefundamental frequency may be detected by subtracting the fundamentalfrequency using a low phase noise PLL or crystal filter.

In certain cases, the received signal contains most of the power in thelower harmonics and lower intermodulation peaks while most of the highfrequency spatial content is in the higher harmonics and higherintermodulation peaks (where “higher” means n or m is greater than 4 and“lower” means n or m is 4 or less). The magnitude of the harmonics maydepend on the size of the magnetic particles, where larger particlesincrease higher order harmonics. Intermodulation may increases thespectral content of the received signal. In some cases, the totalnormalized signal is larger with intermodulation than without. Incertain cases, the intermodulated point-spread function (PSF) around 2f₀is similar to the PSF without intermodulation. As the harmonic number(n) increases, the magnitude of the PSF decreases and the resolutionincreases.

While in some embodiments the intermodulation products around a singleharmonic (e.g., 2f₀±nf₁) may be received, in other embodiments the RFcoils may be configured to receive multiple frequencies around multipleharmonics (e.g., 2f₀±nf₁, 3f₀±nf₁, 4f₀±nf₁, 5f₀±nf₁). For example, oneor more dual-tuned coils may be used, each configured to receive signalsaround two harmonics. Alternatively, multiple RF coils may be separatelytuned to receive the signals around each harmonic.

As the excitation frequency f₀ increases, the frequency separation ofthe harmonics of the excitation frequency increases. In some instances,increasing the excitation frequency may facilitate coil-to-coilisolation, high-Q coil construction, and noiseless rejection of thefundamental. As f₀ increases, the SAR limit may be approached. Forexample, with f₀=1 MHz and detection at 2f₀=2 MHz, a SAR of 4 W/kg maybe reached at 3 mT_(peak-peak) in a small animal.

In certain embodiments, a high-Q receive coil may facilitate asimplification in construction and optimal noise matching of thereceiver system. The bandwidth requirements may be less due to the lowintermodulation frequency (e.g., f₁=200 Hz) as compared to theexcitation frequency. Typically, detecting N harmonics with aconventional MPI would require a bandwidth of BW=Nf₀. Thus, detectingN=8 harmonics using an excitation frequency f₀=150 kHz would require abandwidth of 1.2 MHz, which may result in sub-optimal matching to thepreamplifier. In contrast, with intermodulation, the bandwidth maydepend on the intermodulation frequency (e.g., f₁=200 Hz) instead of theexcitation frequency (e.g., f₀=150 kHz). In these cases, the receivebandwidth may be less than a typical MPI system.

The receiver coils may be configured to convert the detected magneticsignal from the sample into an electrical signal. In some embodiments,separate x, y, z receiver coils are used to measure signals inorthogonal x, y, z directions, as illustrated in FIGS. 6( a) and 6(b).The cross-sectional view of FIG. 6( a) shows the z coil 600 and y coil602. The perspective view of FIG. 6( b) also shows the z coil 600, ycoil 602, and the x coil 604. The z coil has zero net area and is woundas a gradiometer. The zero net area provides reduced feed through of thefundamental frequency f₀. If transverse RF transmit coils are used, thetransverse receive coils can be geometrically decoupled by placing themsuch that less net area occurs between them. The receive coils may bewound so that they produce a homogeneous magnetic field. In some cases,the transverse receive coils are similarly wound with a gradiometerconfiguration.

FIG. 7( d) is a block diagram of a receive circuit chain according to anembodiment of the present disclosure. Three receive circuits may beused, one for each of the x, y, and z directions. Each receive circuitmay include a high-Q receive coil and balus 792, low amplification andlow noise HPF preamplifier 793 and 794, fundamental block 795, low-passfilter 796, band-pass filter 797, LPF amplifier 798, and feedbackcircuit 799. Band-pass filter 797 may be centered around a harmonic suchas 2f₀, but may also be centered around another higher harmonic or sothat multiple harmonics are allowed to pass. The receive circuit may beconstructed to remove the fundamental signal f₀ and only pass harmonicsmf₀, m>1. The circuits shown herein are illustrative examples andvarious alternative circuits may be designed to perform equivalentfunctions.

In some cases, the preamplifier includes operational amplifiers andhigh-Q tuned traps. The output may include an RC filter or bandpassfilter configured to reduce high frequency noise. Optimal matchingbetween the receive coil and preamplifier may occur when the coilresistance at the receive frequency is substantially equal to the ratioof the preamplifier noise voltage to the preamplifier noise current. Incertain instances, the matching may be achieved using baluns andmatching capacitors and inductors in low-pass, high-pass, and band-passconfigurations.

FIGS. 8( a) and 8(b) illustrate two alternative receive coil circuitsfor providing a dual-tuned receive coil. The circuit of FIG. 8( a)includes receive coil 800, first frequency matching circuit elements802, second frequency matching elements 804, and balun 806. Matchingcircuit elements 802 contain an inductor and capacitor tuned to a firstfrequency. Matching circuit elements 804 contain an inductor andcapacitor tuned to a second frequency. The circuit of FIG. 8( b)includes receive coil 808, first frequency matching circuit elements810, second frequency matching elements 812, and balun 814. Matchingcircuit elements 810 contain a capacitor tuned to a first frequency.Matching circuit elements 812 contain an inductor and capacitor tuned toa second frequency. Additional frequency matching elements may be addedto the circuits of FIG. 8( a) or 8(b) in a similar manner to providematching to additional distinct frequencies for a multi-tuned coil.

In some cases, the receiver is configured to be a wideband receiver. Incertain embodiments, a wideband receiver is included in a deviceconfigured for x-space magnetic particle imaging. For example, thereceiver may be configured to have non-resonant matching. In someinstances, the receiver is configured to have non-resonant matching,such that the receive spectrum has a substantially flat amplitude andphase across a wide bandwidth. For instance, the receiver may beconfigured to have a receive bandwidth ranging from 1 kHz to 100 MHz,such as from 1 kHz to 50 MHz, including from 5 kHz to 25 MHz, or from 10kHz to 10 MHz, or from 10 kHz to 5 MHz, or from 10 kHz to 1 MHz, or from20 kHz to 500 kHz, or from 20 kHz to 250 kHz, or from 20 kHz to 200 kHz.In certain embodiments of x-space MPI, the receiver is configured tohave a wide bandwidth. In these embodiments, the device may beconfigured to include noise matching. In some instances, the deviceincludes a transformers and/or a balun configured to perform noisematching. In certain instances, the receiver includes a pre-amplifier.The pre-amplifier may be configured to have low voltage noise, such as1000 nV/√Hz or less, including 500 nV/√Hz or less, for instance 250nV/√Hz or less, or 100 nV/√Hz or less, or 50 nV/√Hz or less, or 25nV/√Hz or less, or 20 nV/√Hz or less, or nV/√Hz or less, or 10 nV/√Hz orless, or 5 nV/√Hz or less, or 4 nV/√Hz or less, or 3 nV/√Hz or less, or2 nV/√Hz or less, or 1 nV/√Hz or less. For example, the pre-amplifiermay include a junction gate field-effect transistor (JFET), or the like.

Signal to Noise Ratio and Noise Matching Theory

When the receive coil and preamplifier dominate the noise in a receiver,noise matching the receiver to the pickup coil may improve SNR. Optimalmatching may occur when the coil impedance is substantially matched tothe voltage and current noise of the preamplifier. In some instances,assuming optimal matching, the dominant noise source may be the coil. Ahigh-Q receiver coil and bandwidth narrowing may increase preamplifierSNR for a given f₀ by reducing the preamplifier noise figure, the coilnoise for a given volume, and the detection bandwidth. In certainembodiments, because these factors are independent of f₀, the detectionfrequency may be increased to reach body noise dominance withoutincreasing receiver bandwidth. In some instances, the imaging region maybe increased by increasing H_(tot)=H₀+H₁. Certain embodiments of thepresent disclosure thus provide narrowband MPI which increases SNR for afixed SAR.

The signal from the MPI receive coil may be amplified prior todigitization. The amplifier noise may be minimized when the receivercoil is noise matched to the preamplifier. In some cases, noise matchingmay be optimized by minimizing the ratio of output noise to input noise,i.e., the noise gain ratio. The noise gain ratio may be minimized whenthe real coil resistance at resonance is R_(coil)=e_(n)/i_(n), wheree_(n) and i_(n) are, respectively, the voltage and current noiseamplitude per unit bandwidth of the preamplifier.

In certain instances, the MPI device has a noise figure of 10 dB orless, such as 5 dB or less, including 3 dB or less, or 1 dB or less, forinstance 0.75 dB or less, or 0.5 dB or less, or 0.25 dB or less, or 0.1dB or less. In some cases, the noise figure is 50% or less greater thanthe theoretical best case noise figure, such as 40% or less, or 30% orless, or 20% or less, or 15% or less, or 10% or less, or 5% or less, or1% or less greater than the theoretical best case noise figure.

In certain embodiments, the noise figure (NF) is represented by theformula:

${NF} = {10{\log_{10}\left( \frac{e_{n}^{2} + {i_{n}^{2}R_{s}^{2}} + {4k\; T\; R_{s}}}{4k\; T\; R_{s}} \right)}}$

where e_(n) and i_(n) are, respectively, the voltage and current noiseamplitude per unit bandwidth of the preamplifier, R_(s) is the coilimpedance, k is the Boltzmann constant, and T is the temperature. Insome instances, NF is maximized when R_(coil)=e_(n)/i_(n).

Assuming optimal matching, then the dominant noise source may be thecoil. Then the coil noise is:

e _(c)=(4kT _(E) ΔR)^(1/2)

and the resistance can be estimated assuming a long straight cylindricalconductor and the skin effect:

R∝(f₀ρ(T_(c)))^(1/2)

where f₀ is the detection frequency and p(T_(c)) is the resistivity ofthe conductor coil at temperature T_(c).

The received signal in contained in the harmonics of ξ:

ξ∝−∫_(sample)(∂/∂t) B ₁· MdV_(s)∝K(f₀)M₀V_(s)f₀

where K(f₀)≦1 models relaxation effects, M₀ is the ferromagneticmagnetization density in the imaging region, and V_(s) is the volume ofthe imaging region. Then, the SNR after the pre-amplifier is:

${SNR} \propto \frac{F\; {K\left( f_{0} \right)}M_{0}V_{\; s}f_{0}^{3/4}}{\left( {4k\; T_{c}\Delta \; f\; {\rho \left( T_{c} \right)}} \right)^{1/2}}$

where F is the noise factor F=10^(NF/10). Thus, bandwidth narrowing anda high-Q receiver coil may increase SNR, since it enables lowpre-amplifier noise figure, minimum coil noise for a given volume, smalldetection bandwidth, high detection frequency, and ability to increasethe imaging region.

In certain embodiments, the magnetic particle imaging device isconfigured to have a signal to noise ratio (SNR) ranging from 1 to1,000,000, such as from 5 to 750,000, including from 10 to 500,000.

Signal Processor

Aspects of the magnetic particle imaging device include a signalprocessor. The signal processor may be configured to convert the signaldetected by the receiver into an image of the magnetic particles in thesample. For example, the signal processor may be configured to convertthe detected signal into a one dimensional image of magnetic particlesin the sample. The one dimensional image may represent a partial fieldof view of the magnetic particles in the sample. In some cases, thesignal processor is configured to combine two or more partial field ofview images into the image of the magnetic particles in the sample. Forinstance, two or more partial field of view images may be combined toproduce a two or three dimensional image of the magnetic particles inthe sample.

In certain embodiments, the signal processor is configured to convertthe detected signals into a multi-dimensional image of the magneticparticles in the sample. The multi-dimensional image may be a nativemulti-dimensional image. By “native” is meant that the image isfundamental to the technique, and does not require pre-characterizationof a specific type of nanoparticle in the system. To do this, anembodiment may image using a partial field of view technique. Forexample, the signal processor may be configured to convert the detectedsignals into native 2D partial field of view images of the magneticparticles in the sample. These native 2D images may further be combinedto produce a larger field of view 2D image, or a 3D image of themagnetic particles in the sample. In other embodiments, the signalprocessor is configured to convert the detected signals into native 3Dpartial field of view images of the magnetic particles in the sample.These native 3D images may be further combined to produce a larger fieldof view 3D image of the magnetic particles in the sample.

The signal processor may be any type of processor capable of convertingthe detected signals into an image of the magnetic particles in thesample, and in some instances includes a computer processor programmedto convert the detected signals into an image of the magnetic particlesin the sample. The signal processor may include additional electroniccircuits and devices, such as but not limited to, analog to digitalconverters, digital to analog converters, digital down-converters,combinations thereof, and the like.

In certain instances, the magnetic particle imaging device is configuredto have a field of view ranging from 0.1 mm to 500 cm, such as from 1 mmto 250 cm, including from 5 mm to 100 cm, or from 1 cm to 50 cm, or from1 cm to 25 cm, or from 1 cm to 10 cm.

Resolution

Spatial resolution of the system is the ability to accurately depict twodistinct points of equal intensity in space. In certain embodiments, themagnetic particle imaging device is configured to have a resolutionranging from 1 cm to 1 nm, such as from 0.5 cm to 100 nm, including from1 mm to 1 μm, for instance from 1 mm to 100 μm, or from 1 mm to 500 μm.For example, the magnetic particle imaging device may be configured tohave a resolution of 1 cm or less, such as 0.5 cm or less, including 1mm or less, or 500 μm or less, or 100 μm or less, or 10 μm or less, or 1μm or less, or 500 nm or less, or 100 nm or less.

In certain embodiments, the device is configured to detect magneticparticles in a sample at different resolutions. For instance, the devicemay be configured to detect magnetic particles in a sample at a firstresolution and then subsequently detect the magnetic particles in thesample at a second resolution. In some cases, the first resolution is alower resolution than the second resolution. In other cases, the firstresolution is a higher resolution than the second resolution. In someinstances, the device is configured to analyze the signals detected atdifferent resolutions. For example, the device may be configured tocombine the signals detected at different resolutions. Analyzing thesignals detected at different resolutions may facilitate the recovery oflow frequency information that may have been lost due to filtering ofthe detected signal. For example, combining the signals detected atdifferent resolutions may facilitate the recovery of low frequency imagedata that may have been lost due to high pass filtering of the detectedsignal.

Sensitivity

The sensitivity of a magnetic particle imaging device relates to theminimum amount of magnetic particles in a sample that can be detected bythe device. In certain embodiments, the magnetic particle imaging systemis configured to have a sensitivity ranging from 1 mg to 1 pg, such asfrom 100 μg to 50 pg, including from 10 μg to 0.1 ng, for instance from1 μg to 0.1 ng. In some cases, the magnetic particle imaging device isconfigured to have a sensitivity of 1 mg or less, such as from 100 μg orless, including from 10 μg or less, or 1 μg or less, or 100 ng, or less,or 10 mg or less, or 1 ng or less, or 0.1 ng or less, or 50 pg or less,or 10 pg or less, or 1 pg or less.

Linearity

In certain embodiments, the magnetic particle imaging device isconfigured to produce a linearly varying signal with respect to theconcentration of magnetic particles in the non-saturating magnetic fieldregion. A linearly varying signal may facilitate the determination ofthe relative amount of magnetic particles in a region of the sample ascompared to other regions of the sample. The amount of magneticparticles in a region of the sample may be represented in an image ofthe magnetic particles by the intensity of the signal in that region.For instance, a region with a relatively large amount of magneticparticles may be represented by a brighter area in the image or an areaof a color associated with a larger amount of magnetic particles.

Shift Invariant

In certain embodiments, the magnetic particle imaging device isconfigured to produce signal that is shift invariant. By “shiftinvariant” is meant that the device is configured to produce a linearlyvarying signal with respect to the concentration of the magneticnanoparticles in the non-saturating magnetic field region, whereshifting the position of the non-saturating magnetic field region doesnot change the linearity of the signal. As such, as described above, thedevice may be configured to produce a scanning magnetic field thatpositions the non-saturating magnetic field region in the magneticfield. In some instances, the device is configured to produce a linearlyvarying signal as the position of the non-saturating magnetic field ismoved through the magnetic field.

Gridding

In certain embodiments, the signal processor is configured to correlatethe detected signals to the position of the non-saturating magneticfield region when the signal was acquired. Correlating the detectedsignal to the instantaneous position of the non-saturating magneticfield region may be referred to herein as “gridding”. In certaininstances, the signal processor is configured to grid the detectedsignals as part of producing an x-space image of magnetic particles in asample. For example, the signal processor may be configured to processthe detected signal to ensure the phase linearity, as described above,and then the processor may be configured to grid the received signal tothe instantaneous location of the non-saturating magnetic field region.In some instances, gridding the detected signals to the position of thenon-saturating magnetic field region when the signal was acquiredfacilitates the production of the native image of the magnetic particles(e.g., a 2D or 3D image of the magnetic particles).

Circuitry

The magnetic particle imaging device may also include circuitryassociated with the excitation signal source and the receiver. FIGS. 7(a) and 7(b) are schematic diagrams of the transmit and receive chains,respectively. The excitation circuitry 700 shown in FIG. 7( a) includesan RF power amplifier 702, second harmonic trap 704, high-pass filter706, and high-frequency resonant coil 708. The excitation circuitry 700may also include an LF power amplifier 718, low-pass filter 716, and LFcoil 714 isolated from the sample 710 and RF chain by RF shield 712. TheRF excitation and LF intermodulation signals are generated by thesechains to excite magnetic particles in sample 710. In certain cases, thesignal generators are phase locked to a coherent detector. In someinstances, the RF amplifier 702 drives the matched, water-cooledresonant transmit coil 708 to generate a sinusoidal magnetic field atf₀=150 kHz with peak-to-peak amplitude of 6 mT. In certain cases, theintermodulation coil 714, driven by audio amplifier 718, generates asinusoidal magnetic field at f₀=200 Hz with peak-to-peak amplitude of5.9 mT.

The receive circuitry 750 shown in FIG. 7( b) includes a receive coil752, balun 754, first preamplifier stage 756, first fundamental trap758, second preamplifier stage 760, second fundamental trap 762, andpreamplifier stage 764. In certain embodiments, the interfacingelectronics are designed to prevent intermodulation in the RF and LFamplifier output stages (FIG. 7( a)) and in the preamplifier (FIG. 7(b)) through the use of high-pass, low-pass, and notch filters. The RFtransmit chain 700 may have a resonant transmit coil 708 at 150 kHz anda second harmonic trap 704 at 300 kHz. The receive chain 750 may have afundamental harmonic trap 758 at 150 kHz and high-Q receive coil 752 at300 kHz. The transfer functions for the transmit and receive chains areshown, respectively, in FIGS. 9( a) and 9(b), which show 80 dBV or moreof isolation between the transmit and receive frequencies. In somecases, each isolation stage, including RF shield, provides approximately30 dB of isolation. In certain embodiments, the inductors are high-Q RFtoroids or are air core. High power capacitors may be used in thetransmit chain and high-Q capacitors may be used in the receive chain.

FIG. 7( c) is a schematic diagram of the transmit chain according tocertain embodiments of the present disclosure. The RF transmit chainincludes RF amplifier 780, 2f₀ filter 782, matching capacitor 784,high-power capacitor 786, RF coil 788, and large capacitor 790. In somecases, the filter 782 reduces the effect of harmonics in the outputspectrum of the RF amplifier. The filter circuit elements inductors L₁,L₂, and capacitor C₁ resonate so that the fundamental frequency f₀passes through the filter. Inductor L₂ and capacitor C₁ may be tuned sothat there is a shunt to ground at 2f₀. Additional shunts at 3f₀, 4f₀,and higher may also be included, up to shunt at Nf₀ provided by L_(N−1)and capacitor C_(N).

In some instances, the preamplifier uses low-noise op-amps in stages756, 760, 764 matched to a high-Q coil 752 of Z_(coil)=2 kΩ atresonance. The preamplifier may be noise matched to the receive coilusing a 4:1 balanced-to-unbalanced impedance transformer (balun) 754.Some embodiments may include feedback circuitry configured to preventphase and magnitude drift in the RF transmit power and excitation fieldstrength caused by loading of the transmit coil and heating. Thefeedback circuit may include a Cartesian feedback circuit or a currentfeedback circuit. The current or field can be measured in various wayssuch as using a pickup coil or a current sensor/shunt resistor. In someinstances, the feedback circuitry facilitates the regulation of thefield that the sample experiences, and the current through the transmitcoil may depend on the feedback circuitry. In some embodiments, feedbackdamping is used to widen the bandwidth of the receive coils. Feedbackdamping may be performed by feeding back the received signal out ofphase to the receive coil, as shown in FIG. 7( d). In some embodiments,a rotating (quadrature) excitation field is used to increase thedetectable signal.

In other embodiments, the device includes a wideband receiver, asdescribed above. The wideband receiver may be configured to have asubstantially flat amplitude and phase across the receive spectrum. Incertain instances, the receiver is configured to include a high passfilter. In these embodiments, the received signal may be filtered by thehigh pass filter and the device may be configured to correct thefiltered signal. In some instances, the filtered signal is corrected toproduce a signal with a substantially flat amplitude and phase, asdescribed above.

Signal Harmonics and Sideband Tones

FIG. 3( a) is a graph of received signal level versus frequency for aconventional MPI technique showing multiple received harmonics such assecond harmonic (2f₀) 302. The graph shows a simulated signal receivedby an untuned pickup coil. In conventional MPI techniques, thefundamental (f₀) 300 is typically not used because it may becontaminated by direct feed through from the excitation field.

FIG. 3( b) is a graph of received signal level versus frequency for anMPI technique using intermodulation according to embodiments of thepresent disclosure. The graph shows a simulated signal envelope 304received by an untuned pickup coil. Intermodulation generates detectableintermodulation products around the main harmonics, spreading thespectral information. These sideband intermodulation products around themain harmonics are absent in conventional MPI techniques that do not useintermodulation. In some cases, the amplitude and phase of theintermodulation products vary as a function of position. Each peak inthe envelope may contain multiple intermodulation sideband tones. Forexample, details of the tones contained in peak 306 around the secondharmonic are shown in FIG. 3( c). Tone 308 is the second harmonic (2f₀).Clustered around this harmonic are intermodulation tones such as tone310 at frequency 2f₀+f₁, tone 312 at frequency 2f₀+2f₁, and tone 314 atfrequency 2f₀+3f₁. Because f₀ is significantly larger than f₁, thebandwidth required to receive the seven harmonics 2f₀ through 8f₀ inFIG. 3( a) is larger than the bandwidth required to receive the sevenintermodulation tones 2f₀-3f₁ through 2f₀+3f₁. As a result,intermodulation allows the use of a narrowband receiver to detect asimilar number of distinct frequency signals. This, in turn, allows theuse of a tuned receiver and improved signal to noise ratio (SNR).

Magnetic Particles

Any convenient magnetic particle may be employed. Magnetic particles aremagnetizable particles that, when exposed to an excitation signal, aredetectable by the magnetic particle imaging device. Magnetic particlesof interest may be configured to be substantially saturated in themagnetic field produced by the magnetic source of the magnetic imagingdevice, but not saturated in the non-saturating magnetic field region ofthe magnetic field.

Magnetic particles useful in the practice of certain embodiments of thepresent disclosure are magnetic (e.g., ferromagnetic) colloidalmaterials and particles. The magnetic nanoparticles can be high momentmagnetic nanoparticles which may be super-paramagnetic, or syntheticanti-ferromagnetic nanoparticles which include two or more layers ofanti-ferromagnetically-coupled high moment ferro-magnets. Both types ofmagnetic particles appear “nonmagnetic” (e.g., have a magnetization ofsubstantially zero) in the absence of a magnetic field. Magneticparticles with a substantially zero remnant magnetization may notsubstantially agglomerate with each other in solution in the absence ofan external magnetic field. In accordance with certain embodiments,magnetizable particles suitable for use include one or more materialssuch as, but not limited to, paramagnetic, super-paramagnetic,ferromagnetic, and ferri-magnetic materials, as well as combinationsthereof, and the like.

In certain embodiments, the magnetic particles have remnantmagnetizations that are small or substantially zero, such that they willnot agglomerate in solution. Examples of magnetic particles that havesmall remnant magnetizations include super-paramagnetic particles andanti-ferromagnetic particles. In certain cases, the magnetic particleshave detectable magnetic moments under a magnetic field of 1 T or less,such as 100 mT or less, or 10 mT or less, or 1 mT or less. In someembodiments, the magnetic labels are colloidally stable, e.g.,nanoparticle compositions may be present as a stable colloid. Bycolloidally stable is meant that the nanoparticles are evenly dispersedin solution, such that the nanoparticles do not substantiallyagglomerate. In certain embodiments, to prevent clumping, thenanoparticles may have no net magnetic moment (or a very small magneticmoment) in zero applied field.

The magnetic particles may be chemically stable in a biologicalenvironment, which may facilitate their use in the assay conditions. Insome cases, the magnetic particles are biocompatible, i.e., watersoluble and functionalized so that they may be readily attached tobiomolecules of interest, e.g., a antibody that specifically binds to atarget analyte. By associating or binding magnetic particles to aspecific antibody, the magnetic particles may be targeted to specificareas of the body in a subject through the specific binding interactionsbetween the antibody and complementary antigen. In some instances, themagnetic label may be bound to the protein or antibody as describedabove through a non-covalent or a covalent bond with each other.Examples of non-covalent associations include non-specific adsorption,binding based on electrostatic (e.g. ion, ion pair interactions),hydrophobic interactions, hydrogen bonding interactions, specificbinding through a specific binding pair member covalently attached tothe surface of the magnetic particle, and the like. Examples of covalentbinding include covalent bonds formed between the biomolecule and afunctional group present on the surface of the magnetic particle, e.g.—OH, where the functional group may be naturally occurring or present asa member of an introduced linking group.

In certain embodiments, the magnetic nanoparticles are high momentmagnetic particles such as Co, Fe or CoFe crystals, which may besuper-paramagnetic at room temperature. Magnetic nanoparticles suitablefor use herein include, but are not limited to, Co, Co alloys, ferrites,cobalt nitride, cobalt oxide, Co—Pd, Co—Pt, iron, iron alloys, Fe—Au,Fe—Cr, Fe—N, Fe₃O₄, Fe—Pd, Fe—Pt, Fe—Zr—Nb—B, Mn—N, Nd—Fe—B,Nd—Fe—B—Nb—Cu, Ni, Ni alloys, combinations thereof, and the like. Insome embodiments, a functionalized coating, such as, but not limited to,a thin layer of gold plated onto a magnetic core, or a poly-L-lysinecoated glass surface, may be attached to the magnetic core. These typesof coating may facilitate the attachment of biomolecules, such asantibodies, antigens, proteins, etc. to the surface of the magneticparticles. In some embodiments, to facilitate the bio-conjugation of thenanoparticle, a gold cap (or cap of functionally analogous or equivalentmaterial) is layered on the top of the layers of anti-ferromagneticmaterial so that the nanoparticle can be conjugated to biomolecules viaa gold-thiol or other convenient linkage. Surfactants may be applied tothe nanoparticles, such that the nanoparticles may be water-soluble. Theedges of the nanoparticles can also be passivated with Au or other inertlayers for chemical stability.

In some cases, the magnetic particles may include two or moreferromagnetic layers, such as Fe_(x)CO_(1-x), where x is 0.5 to 0.7, orFe_(x)CO_(1-x) based alloys. These ferromagnetic layers may be separatedby nonmagnetic spacer layers such as Ru, Cr, Au, etc., or alloysthereof. In certain cases, the spacer layers include ferromagneticlayers coupled antiferromagnetically so that the net remnantmagnetization of the resulting particles are substantially zero. In somecases, the antiferromagnetic coupling strength is such that theparticles can be saturated (i.e., magnetization of all layers becomeparallel) by an external magnetic field of 10 mT or less. In some cases,the antiferromagnetic coupling strength depends of the layer thicknessesand the alloy composition of the spacer layer.

In certain embodiments, the magnetic particles are nanoparticles. By“nanoparticle” is meant a particle having an average size (e.g.,diameter) in the range of 1 nm to 1000 nm. In certain embodiments, thesize (e.g., mean diameter) of the magnetic nanoparticles is sub-micronsized, e.g., from 1 nm to 1000 nm, or from 1 nm to 500 nm, or from 5 nmto 250 nm, such as from 5 nm to 150 nm, including from 5 nm to 50 nm.For example, magnetic nanoparticles having a mean diameter of 5 nm, 6nm, 7 nm, 8 nm, 9 nm, 10 nm, 11 nm, 12 nm, 13 nm, 14 nm, 15 nm, 16 nm,17 nm, 18 nm, 19 nm, 20 nm, 25 nm, 30 nm, 35 nm, 40 nm, 45 nm, 50 nm, 55nm, 60 nm, 70 nm, 80 nm, 90 nm, 100 nm, 110 nm, 120 nm, 130 nm, 140 nm,150 nm, and 200 nm as well as nanoparticles having mean diameters inranges between any two of these values, are suitable for use herein. Incertain embodiments, the magnetic particles are substantially uniform inshape. For example, the magnetic particles may be spherical in shape. Inaddition to a spherical shape, magnetic nanoparticles suitable for useherein can be shaped as disks, rods, coils, fibers, pyramids, and thelike.

In certain embodiments, the magnetic particles include two or moremagnetic sub-particles associated with each other. The associatedmagnetic sub-particles may be coated with a coating to form the magneticparticle. In some cases, the coating is a polymer, such as, but notlimited to, dextran, carboxydextran, and the like. In some cases, themagnetic particles include Resovist® super-paramagnetic iron oxide(SPIO) nanoparticles (Bayer-Schering).

An example of an MPI apparatus according to an embodiment of the presentdisclosure is shown in FIG. 10( a). NdFeB ring magnets 1006, 1007 createa static inhomogeneous magnetic gradient field having a field-freeregion located near the center of imaging bore 1008. Ring magnets 1006,1007 have a mean diameter of 7.62 cm and a center-to-center separationof 6.85 cm. The magnetic field is approximately linear axially down thebore, with a gradient of dB/dz=4.5 T/m. Coronal gradients aredB/dx=dB/dy=2.6 T/m. Water-cooled excitation solenoid 1014 generates adynamic magnetic field that is superimposed on the static field and canexcite magnetic particles in the imaging bore 1008. In addition,intermodulation solenoid 1002 generates a dynamic magnetic field that isalso superimposed on the static field. Magnetic shield 1000 passivelyisolates the AC excitation solenoids 1002 and 1014 from interaction withother components to reduce unwanted heating and signal interference.Signals from magnetic particles located in the imaging bore 1008 arereceived by concentric gradiometer receive coil 1012. The mechanicalframe for the apparatus includes a G10 plate 1010 for mounting ringmagnets 1006, 1007 and aluminum bolt 1004.

FIG. 10( b) is a perspective cut-away view of a device according to anembodiment of the present disclosure. The device contains NdFeBpermanent magnets 1020, 1022 that produce an inhomogeneous field with afield-free point in the center of the bore 1024, water cooledelectromagnet coils (not shown) positioned inside bore 1024 to generatea radio-frequency excitation magnetic field along the longitudinalz-axis and water cooled electromagnet coils to generate low-frequencyexcitation magnetic fields along the x-axis 1026, y-axis 1028, 1030, andz-axis. The x, y, and z axis electromagnets also generate a scanningmagnetic field to move the field-free-point.

The permanent magnets produce a z-gradient of 8 T/m and x-y-gradient of4 T/m. In one implementation, the inner diameter is 3 inches and theouter diameter is 12 inches, and the interior is potted with epoxy toreduce vibration. The RF excitation coils (not shown) positioned insidebore 1024 are driven by a 750 Watt continuous power amplifier. The LFcoils are driven by an 800 Amp peak-to-peak amplifier.

Methods

Aspects of the present disclosure also include a method of imagingmagnetic particles in a sample. The method includes applying a magneticfield having a non-saturating magnetic field region (e.g., a field freeregion, FFR) to a sample that includes magnetic particles. As describedabove, the magnetic particle imaging device includes magnetic fieldsources that produce a saturating magnetic field of sufficient strengththat magnetic particles in the magnetic field are saturated, except forthose magnetic particles that are in the non-saturating magnetic fieldregion of the magnetic field. Application of the magnetic field to thesample may saturate the magnetic particles in the magnetic field and maynot saturate the magnetic particles in the non-saturating magnetic fieldregion.

The magnetic particles in the non-saturating magnetic field region maybe excited by an excitation signal. As such, in certain embodiments, themethod includes applying an excitation signal to the magnetic particlesin the non-saturating magnetic field region to produce a detectablesignal from the magnetic particles in the non-saturating magnetic fieldregion. Applying the excitation signal may include applying two or moreexcitation signals to the magnetic particles in the non-saturatingmagnetic field region. For example, the applying may include applying anRF excitation signal to the magnetic particles in the non-saturatingmagnetic field region. In certain instances, the applying may includeapplying an intermodulation signal, such as a low frequency (LF)intermodulation signal to the magnetic particles in the non-saturatingmagnetic field region.

The excitation signal sources may be configured in various orientationsand spatial arrangements. In some embodiments, the excitation signalsource is configured to create a transverse field along the length ofthe imaging area of the magnetic particle imaging device. In someembodiments, intermodulation may be applied separately and sequentiallyto the RF excitation field in the x, y, and z directions. For example,intermodulation may be applied in the x direction alone, while nointermodulation is used in either the y or z directions. In some cases,intermodulation is applied sequentially to just the y direction, andthen just to the z direction. The intermodulation may also be applied ina combination of directions at once, e.g., a rotating x-y field createdby phase-shifted x and y intermodulation waveforms.

In certain embodiments, the method of imaging magnetic particles in thesample also includes detecting a signal from the magnetic particles inthe non-saturating magnetic field region. In some cases, the detectingincludes detecting the signals from the magnetic particles using one ormore receivers.

Methods of the present disclosure further include analyzing the signalto produce an image of the magnetic particles in the sample. Theanalyzing may include converting the detected signals into one or morepartial field of view images of the magnetic particles in the sample. Insome cases, the analyzing further includes combining the partial fieldof view images into the image of the magnetic particles in the sample.For example, the analyzing may include converting the detected signalsinto a one dimensional image that represents a partial field of view ofthe magnetic particles in the sample. The method may further includecombining two or more one dimensional partial field of view images intoa two or three dimensional image of the magnetic particles in thesample. In some instances, combining the partial field of view imagesinto the image of the magnetic particles in the sample is performed aspart of an x-space magnetic particle imaging method.

In certain embodiments, the analyzing includes converting the detectedsignals into a multi-dimensional image of the magnetic particles in thesample. The analyzing may produce native multi-dimensional images, suchas native 2D or 3D images. For example, the analyzing may includeconverting the detected signals into a native 2D partial field of viewimage of the magnetic particles in the sample. The analyzing may furtherinclude combining two or more of the native 2D images to produce alarger field of view 2D image or a 3D image of the magnetic particles inthe sample. In some instances, the analyzing includes converting thedetected signals into a native 3D image, such as a native 3D partialfield of view image of the magnetic particles in the sample. Theanalyzing may further include combining the 3D partial field of viewimages into a larger field of view 3D image of the magnetic particles inthe sample. In certain embodiments, combining two or more partial fieldof view images into a larger field of view image facilitates therecovery of low frequency information (e.g., low frequency image data)in the detected signal that may be lost during filtering (e.g., highpass filtering) of the detected signal.

In certain embodiments, the analyzing includes correlating the detectedsignals to the position of the non-saturating magnetic field region whenthe signal was acquired. Correlating the detected signal to theinstantaneous position of the non-saturating magnetic field region maybe referred to herein as “gridding”. For example, the analyzing mayinclude gridding the received signal to the instantaneous location ofthe non-saturating magnetic field region. In some instances, griddingthe detected signals to the position of the non-saturating magneticfield region when the signal was acquired facilitates the production ofthe native image of the magnetic particles (e.g., a 2D or 3D image ofthe magnetic particles). In certain cases, the analyzing includesgridding as part of producing an x-space magnetic particle image.

In certain embodiments, receiver signal processing includes using adual-lock in amplifier system to receive the signal. The signal may bedownmixed centered at two and three times the high-frequency RFexcitation. The downmixed signal may then be downmixed again at theLF*(0, +/−1, +/−2, +/−3, etc.).

In some cases, the signal from the receive coil and preamplifier chainis transmitted to a digital down-converter circuit block thatdown-converts the signal to baseband with channelization. The circuitblock may independently down-sample each intermodulation signal so thateach subband tone is channelized. For example, intermodulation products310, 312, 314 shown in FIG. 3( c) are separately down-converted andsampled. During operation, these intermodulation products may becontinuously sampled and stored, associating the stored signal with acorresponding position of the field-free region.

FIGS. 11( a) and 11(b) are block diagrams of signal processing circuitblocks used to process the signals from the receive coil circuit chains.The circuit block 1106 of FIG. 11( a) receives signals originating fromthe receiver coil 1100 and preamplifier 1102. The signals are digitizedby 14 bit 65 MHz analog-to-digital converter 1104 and then separatedinto in-phase and quadrature signal components and down-sampled toproduce corresponding components I(t) and Q(t) of digital signal 1114,s_(m=2)(t). This signal may have a bandwidth of 30 kHz to 100 kHz ormore. Block 1106 also contains microcontroller 1108 which provides x(t),y(t), z(t) control signals to 3D sample control block 1112 in order tocontrol relative 3D translation between the sample and the field-freeregion. Digitized signal 1114 is fed into processing block 1116 of FIG.11( b) where its intermodulation products are independently channelizedand down-sampled to produce in-phase and quadrature component signalsI(t) and Q(t) for each of N subbands around the harmonic. These Nsubband signals are then stored according to associated x(t), y(t), z(t)position coordinates 1120 in N corresponding image memory blocksh₁(x,y,z), . . . , h_(N)(x,y,z), such as block 1118. The processingblocks of FIG. 11( b) may be implemented by a GPU accelerated computeror using a field programmable gate array (FPGA) or application-specificintegrated circuit (ASIC).

In certain embodiments, producing the image of the magnetic particles inthe sample depends on the magnetic particle and the field-free point,which corresponds to a point-spread function (PSF) that depends onvarious factors such as the particle size, magnetic field gradient, andintermodulation product. In some instances, the measured PSF has higherSNR in sidebands closer to the harmonic (e.g., 2f₀ and 2f₀±f₁), butcontains more high frequency spectral content in the sidebands furtherfrom the harmonic (e.g., 2f₀±2f₁, 2f₀±3f₁, 2f₀±4f₁, etc.). In someembodiments, the high SNR of the lower sidebands may be combined withthe higher resolution of the upper sidebands to form a reconstructed(e.g., combined) image. In the reconstructed image, if the uppersidebands become unavailable as SNR drops, then the image resolution maydecrease.

In certain embodiments, image reconstruction includes processing thedetected signals. For example, the detected signals from differentharmonics and/or intermodulation sidebands may be separatelydown-converted and stored to form a set of N distinct images h₁(x,y,z),. . . , h_(N)(x,y,z), each corresponding to a different frequency. Eachof these images may be a convolution of the unknown magnetization. Animage may be reconstructed by a method of parallel deconvolution ofthese images to form a single composite image. In some cases, a Fouriertransform is applied to each of the detected signals s_(n)(x,y,z) toobtain a frequency-domain representation, Y_(n)(k)=F[s_(n)(x,y,z)],where k is a vector indexing the frequency domain and F is the Fouriertransform. If H_(n)(k) represents the n-th harmonic point spreadfunction (PSF) of a point source (which can be determined by calibrationusing a magnetic particle smaller than the system's resolvable limit),and M(k) is the unknown magnetization distribution in the frequencydomain (i.e., the Fourier transform of the unknown magnetizationdistribution in the spatial domain), then Y_(n)(k)=H_(n)(k)M(k)+N_(n)(k), where N_(n)(k) is the Fourier transform of the unknownn-th harmonic noise image. Thus, in some instances, finding the desiredM(k) is equivalent to finding the slope of a complex line given theregression data {Y_(n)(k),H_(n)(k)}. There are many ways of finding theslope, e.g., using a least-squares fit. The reconstructed compositeimage in the frequency domain may then be given byM(k)=H^(T)*(k)·Y(k)/(H^(T)*(k) H(k)), where H(k) and Y(k) areN-dimensional column vectors whose components are H_(n)(k) and Y_(n)(k),respectively. The reconstructed composite image is thenm(x,y,z)=F[M(k)]. This method solves for each frequency-domain pointseparately. Because the least-squares problem increases linearly withthe number of points, it is not the computationally limiting factor, asthe fast Fourier transform (FFT) used to prepare the data scales withO(N·log N). FIG. 12 illustrates the parallel deconvolution of multiplefrequency images 1200, 1202, 1204 to produce a composite image 1206.

The reconstruction method described above may amplify noise athigher-frequency points in the frequency domain where SNR in thereference images are low. Accordingly, embodiments of the presentdisclosure provide methods to address this increase in noise athigher-frequencies. A non-linear processing step may be used to degradereconstructed image resolution while improving the composite image.Specifically, points in the frequency domain with insufficient SNR maybe set to zero:

${M(k)} = \left\{ \begin{matrix}0 & {{\sum\limits_{n = 1}^{N}{{H_{n}(k)}}} < ɛ} \\{{H^{T*}(k)} \cdot {{Y(k)}/\left( {{H^{T*}(k)} \cdot {H(k)}} \right)}} & {otherwise}\end{matrix} \right.$

where ε is an experimentally determined threshold that depends on theSNR. In an alternative thresholding technique, each element where|H_(n)(k)|>ε is used and the others are removed, as follows:

${G_{n}(k)} = \left\{ {{\begin{matrix}0 & {{{H_{n}(k)}} < ɛ} \\{H_{n}(k)} & {otherwise}\end{matrix}{M(k)}} = \left\{ \begin{matrix}0 & {{\sum\limits_{n = 1}^{N}{{G_{n}(k)}}} = 0} \\{{G^{T*}(k)} \cdot {{Y(k)}/\left( {{G^{T*}(k)} \cdot {G(k)}} \right)}} & {otherwise}\end{matrix} \right.} \right.$

The parallel deconvolution technique described here can theoreticallyincrease the SNR of the composite image by √N, but may provide less gainat high spatial frequencies, where fewer harmonics are used in the imagereconstruction. For regions of k-space where none of the N harmonics orintermodulation terms satisfy the condition |H_(n)(k)|>ε, then thisregion of k-space M(k) may be set to zero. In certain cases, m(x,y,z) iscomputed by an inverse FFT algorithm using a programmed processor (e.g.,a computer).

In certain embodiments, the method further includes repositioning thenon-saturating magnetic field region relative to its initial position inthe sample. The non-saturating magnetic field region may be repositionedrelative to its initial position in the sample by repositioning thenon-saturating magnetic field region relative to the sample,repositioning the sample relative to the non-saturating magnetic fieldregion, or a combination of the above. For example, the non-saturatingmagnetic field region may be repositioned by applying a scanningmagnetic field to the magnetic field having the non-saturating magneticfield region. The scanning magnetic field may interact with the magneticfield lines of the magnetic field causing them to deflect from theiroriginal vectors, thus displacing the non-saturating magnetic fieldregion from its initial position. In some instances, the scanningmagnetic field may be applied along one or more axes, such as, but notlimited to the x, y and/or z axes, where the z-axis is aligned with thelongitudinal axis of the magnetic imaging device.

In certain embodiments, the non-saturating magnetic field region may berepositioned relative to the sample by linear translation of the samplewithin the magnetic field. For example, the sample may be attached to asubstrate (e.g., a sample stage), where the substrate is configured tobe movable in one or more directions, such as, but not limited to,directions along the x, y and/or z axes, as described above. In someinstances, by combining methods that include applying a scanningmagnetic field to reposition the non-saturating magnetic field regionand methods that include linear translation of the sample within themagnetic field, a larger field of view may be obtained than by the useof either method alone.

In some instances, the method further includes: repositioning thenon-saturating magnetic field region in the magnetic field; andrepeating the detecting and repositioning to detect a plurality ofsignals from the magnetic particles in the non-saturating magnetic fieldregion. As described above, the plurality of signals may be analyzed andcombined to produce the image of the magnetic particles in the sample.

Embodiments of the method provide techniques for adaptive scanning toimprove efficiency. According to an embodiment of adaptive scanning, aninitial scan (e.g., a scout scan) is performed at lower resolution. Thelow resolution scan may be performed relatively quickly by optimizingsignal acquisition to just the first and/or second and/or thirdharmonics, which would require less time to acquire than a highresolution scan. The low resolution scan can sample points in theimaging region at a lower spatial density as compared to a scan usingthe maximum resolution of the magnetic imaging device. For example,FIGS. 13( a) and 13(b) illustrate a technique of adaptivemulti-resolution scanning. The imaging region 1300 shown in FIG. 13( a)is scanned quickly at a low resolution, sampling at each of the samplepoints such as point 1302. A subset of the sample points have a detectedsignal, such as sample point 1304. The resulting image is then analyzedto identify within the imaging region 1300 an approximate perimeter 1306containing sample points such as point 1304 where the magnetic particleswere detected. Using this information, a higher resolution scan can beperformed within the identified perimeter, as shown in FIG. 13( b). Thespatial density of sample points, such as point 1308, may be higher thanin the low resolution scan. In certain cases, the adaptive scanningtechnique can reduce total scan time because the low resolution scan canbe performed relatively quickly and is then used to eliminate time thatwould otherwise be wasted during a slower high resolution scan of theentire imaging region. In certain instances, adaptive scanning isperformed as part of a method for imaging magnetic particles in a sampleusing magnetic particle imaging device that includes a narrowbandreceiver. In other instances, adaptive scanning is performed as part ofa method for imaging magnetic particles in a sample using magneticparticle imaging device that includes a wideband receiver (e.g., in anx-space magnetic particle imaging device).

In some cases, scanning includes using a dynamic gradient reductiontechnique which involves dynamically changing the gradient strength. Incertain instances, the method includes varying the magnetic fieldstrength of the magnetic field while applying the scanning magneticfield. For example, the magnetic field strength of the magnetic fieldmay be varied as the scanning magnetic field is applied to the magneticfield to reposition the non-saturating magnetic field region. In certaininstances, varying the magnetic field strength while applying thescanning magnetic field allows different resolution images to beproduced by the magnetic particle imaging device. In some cases,acquiring images having different resolutions facilitates the recoveryof low frequency information in the detected signal that may be lostduring filtering of the detected signal. For example, combining thesignals detected at different resolutions may facilitate the recovery oflow frequency image data that may have been lost due to high passfiltering of the detected signal. In certain embodiments, the methodincludes both combining two or more partial field of view images into alarger field of view image, as described above, and varying the magneticfield strength of the magnetic field while applying the scanningmagnetic field.

Scanning can also include modifying the waveform and amplitude of theintermodulation field, and determining the acquisition trajectory of thenon-saturating field free region as it traverses through the sample. Insome cases, the acquisition time may be minimized by determining amathematically optimal acquisition trajectory that may be modified inreal time, i.e., as more data is acquired, the system determines whichpart of the sample to scan for more signal and where to refine.

As an alternative to the two-step adaptive scanning technique describedabove, the acquisition of low and high resolution signals may beperformed on a point-by-point basis during one scan. Specifically, theimaging region may be scanned as follows. At each coarse low-resolutionsample point a signal is acquired, as described above in relation toFIG. 13( a). The coarse sampling may be performed at a resolution lessthan the maximum resolution of the magnetic imaging device. Beforeproceeding to scan the next coarse data point, the low-resolutionsignals at the current coarse data point are examined to determine ifthere is a significant detected signal, which may indicate the presenceof magnetic particles in the area (e.g., “super-pixel”) surrounding thecoarse sample point. If a signal is detected, then a high-resolutionsampling of the area (“super-pixel”) surrounding the coarse point isthen performed and the signals are acquired at a high resolutionsampling structure. The fine or high-resolution sampling is finer orcomparable to the resolution of the highest harmonic or intermodulationimage. The scanning then proceeds to the next coarse sample point andrepeats the process until the entire imaging region is scanned. Morecomplex scanning schemes than those described above could also beemployed. For example, the number of signals, N, at the center of thesuper-pixel could be used to vary the instantaneous sampling distancefrom only one sample per super-pixel to fully sampled at the finestresolution of the highest harmonic or intermodulation image.

The non-saturating magnetic field region (i.e., the field-free region)allows magnetic particles in the region to be detected at once.Electromagnets or physical translation may be used to shift thenon-saturating magnetic field region relative to the sample. Inaddition, the non-saturating magnetic field region may be rotatedrelative to the sample. Rotation of the non-saturating magnetic fieldregion relative to the sample, e.g., by geometric rotation of thepermanent magnets and/or the scanning magnetic sources relative to thesample, allows computed tomography techniques to be used for imagereconstruction. For example, FIGS. 2( b) and 2(c) show a sample 204being imaged using such a technique. In FIG. 2( b) the non-saturatingmagnetic field region 208 (in this case, in the form of a line, e.g., afield free line) is displaced in a direction perpendicular to the lineto various other positions such as 206. At each position of the line,data is acquired. The data from all the line positions is then used toform an image slice. The orientation of the field-free line relative tothe sample 204 is then changed, as shown in FIG. 2( c), by rotating thepermanent magnets together with the scanning magnetic sources relativeto the sample. The field-free line is then displaced again to variouspositions such as 210 and 212, and data is again acquired at eachposition of the field-free line to form an image slice. Thus, acollection of image slices are acquired, each having a unique angleassociated with it. For example, image slices may be acquired for acollection of distinct angles ranging uniformly over 180 degrees.Computed tomographic techniques are then used to generate an image ofthe sample from the collection of image slices. The field-free lineallows for a projection format for MPI, which in some cases decreasesthe time required to image a sample.

Aspects of the present disclosure also include a method of producing animage of magnetic particles in a subject. The method includesadministering magnetic particles to a subject. The magnetic particlesmay be administered to the subject by various methods, such as, but notlimited to, ingestion, injection, inhalation, combinations thereof, andthe like. For example, the magnetic particles may be provided in aninjectable solution and the magnetic particles may be administered tothe subject by injecting the injectable solution including the magneticparticles into the subject. The magnetic particles may diffuse throughthe area surrounding the injection site, or if injected into a bloodvessel, may be carried through the blood vessel to an imaging site. Insome instances, the magnetic particles are non-specific such that themagnetic particles freely diffuse through the body and do not targetspecific tissues in the subject. In some case, the magnetic particlesmay be associated with a targeting moiety, such as an antibody, whichspecifically associates with a complementary antigen. Associating amagnetic particle with an antibody may facilitate targeting the magneticparticles to specific sites in the subject base on the specificity ofthe antibody-antigen interactions.

For example, in medical imaging applications, magnetic nanoparticles maybe components of a contrast agent that may be distributed in a subject,e.g., by injection into an organism or labeled into or onto cells. Thesubject, which may be animate or inanimate, human, animal, or otherorganism or portion thereof, is then positioned into the apparatus forimaging. To detect the concentration of magnetic particles in differentregions, the field-free region is scanned relative to the subject byphysical movement of the subject relative to the apparatus and/ordisplacement of the field-free region by dynamically changing themagnetic field using a scanning magnetic field as described above. Forexample, movement of the field-free region can be produced by acombination of physical translation in the axial direction with dynamicscanning in the transverse plane. The scanning can also be produced byphysical translation alone or dynamic scanning alone.

At the field-free region, one or more oscillating excitation magneticfields may be used to excite the magnetic particles situated in thefield-free region. These oscillating excitation fields may haveamplitudes in the range of 0.1 mT to 30 mT, as described above. Theexcitation fields cause the magnetization of the particles to saturate,generating harmonics that can be isolated from the fundamental usingfrequency domain techniques. The harmonic response at each field-freeregion is detected using one or more receive coils, and the detectedsignals are analyzed at each point to create a complete scan of thedistribution of particles in the imaging region.

Utility

Magnetic particle imaging (MPI) in accordance with the embodiments ofthe present disclosure finds use in various applications, e.g., where itis desired to determine the distribution of magnetic particles in asample. For example, the distribution of magnetic particle in a samplemay be used to visualize the internal structure of the sample, where themagnetic particles are concentrated in certain areas within the sample.The magnetic particles may be administered to a subject, for instance byinjection, and the subsequent position of the magnetic particles in thesubject may be determined by MPI.

In some instances, MPI is used for the imaging of blood vessels. MPI maybe used in an angiography method that may produce an image of themagnetic particles and not see tissue. Current techniques forangiography, fluoroscopy, CT Angiography (CTA), and MR Angiography(MRA), inherently see tissue in addition to the tracer agent. As aresult, fluoroscopy typically requires high concentrations of iodinetracer and high resolutions are required in CTA and MRA to differentiatethe tracer from surrounding tissues. Since MPI only detects the magnetictracer, the MPI image of a blood vessel filled with magnetic particletracer would be an image of the blood vessel convolved with the pointspread function. A stenosis, for example, would be seen as darkening ofthe blood vessel even if the diameter of the blood vessel was below theintrinsic resolution of the image.

In some instances, Magnetic Particle Imaging (MPI) in accordance withembodiments of the present disclosure finds use as a medical imagingtracer modality with potential applications in human or small animalangiography, cancer imaging, in vivo cell tracking and inflammationimaging. In certain embodiments, MPI is a linear shift-invariant imagingsystem with an analytic point spread function. Some aspects of thepresent disclosure include a fast image reconstruction method thatobtains the intrinsic MPI image with high SNR via gridding the detectedsignals in x-space. In some instances, methods are provided toreconstruct large field of view (FOV) images using partial field-of-viewscanning, despite the loss of first harmonic image information due todirect feedthrough contamination. In some cases, aspects include amulti-dimensional x-space MPI. For example, MPI in accordance withembodiments of the present disclosure find use in cell imaging and celltracking, such as cancer cell imaging. Magnetic particles may beassociated with a specific binding pair member (e.g., an antibody) thatspecifically binds to an antigen (e.g., an antigen expressed on cancercells). After administration of the magnetically labeled specificbinding pair member to a subject, the subject may have an MPI imagetaken of various areas within the subject to determine the location ofthe magnetic particles in the subject. The location of the magneticallylabeled particles in the subject may be indicative of specific bindingbetween the specific binding pair member (e.g., the antibody) and itscomplementary target (e.g., an antigen), which in turn may be indicativeof the presence and location of cells that express that antigen in thesubject.

In certain embodiments, MPI in accordance with embodiments of thepresent disclosure uses the nonlinear magnetic characteristics of ironoxide nanoparticles to generate an image whose resolution depends on themagnetic properties of the nanoparticle and the magnitude of thelocalizing magnetic field gradient. In certain embodiments, the spatialresolution of MPI is finer than the wavelength of the electromagneticfields used to interrogate the magnetic nanoparticles. In some cases,MPI uses no ionizing radiation and has high tracer imaging contrastsince there is no background signal from tissue because tissue istransparent to low frequency magnetic fields.

Magnetic particle imaging (MPI) in accordance with the embodiments ofthe present disclosure finds use in applications where time resolvedimaging is desired. By time resolved imaging is meant that signals frommagnetic particles can be acquired and processed into a plurality ofimages over a period of time. As such, the image produced by a magneticparticle imaging device as disclosed herein may be a time resolvedimage, where the image includes a plurality of images of the magneticparticles in the sample over a period of time. Time resolved imaging mayfacilitate the observation of changes in magnetic particle density indifferent areas of a sample over time. For example, angiography mayinvolve the detection of a plurality of magnetic particle images overtime. Analysis of a plurality of magnetic particle images over time mayfacilitate the observation of the flow of the magnetic tracer throughblood vessels over time. Other embodiments of time resolved magneticparticle imaging may find use in the detection and/or tracking of cells(e.g., cancer cells) in a subject. For example, after administration ofa magnetically labeled specific binding pair member, as described above,a plurality of MPI images may be taken of the subject, and the locationof the magnetically labeled binding pair member may be tracked overtime. Aggregation of the magnetic particles at a particular location inthe subject over time may be indicative of the presence of the specifictarget bound by the magnetically labeled specific binding pair member,and thus may be indicative of the presence of cells that express thatantigen at that location.

Magnetic particle imaging (MPI) in accordance with the embodiments ofthe present disclosure finds use in applications such as interventionalradiology (also known as vascular and interventional radiology, orimage-guided surgery or surgical radiology). Interventional radiologyinvolves the use of minimally invasive procedures performed using imageguidance. Magnetic particle images may be used to direct interventionalprocedures, which are usually done with needles and narrow tubes such ascatheters. The magnetic particle images provide images that may allow aninterventional radiologist to guide instruments through the subject tothe areas containing disease. In certain instances, magnetic particleimaging may minimize physical trauma to the subject by allowing aninterventional radiology procedure to be performed on the subject,rather than a typical surgical procedure. In addition, the use ofmagnetic particle imaging in interventional radiology may facilitate areduction in infection rates and recovery time for the subject.

The following examples are put forth so as to provide those of ordinaryskill in the art with a complete disclosure and description of how tomake and use the embodiments disclosed herein, and are not intended tolimit the scope of what the inventors regard as their invention nor arethey intended to represent that the experiments below are all or theonly experiments performed. Efforts have been made to ensure accuracywith respect to numbers used (e.g., amounts, temperature, etc.) but someexperimental errors and deviations should be accounted for. Unlessindicated otherwise, parts are parts by weight, molecular weight isweight average molecular weight, temperature is in degrees Centigrade,and pressure is at or near atmospheric.

Experimental One-Dimensional X-Space Formulation of the MagneticParticle Imaging Process

MPI can be understood as a x-domain process approached as aone-dimensional system, solving for the point spread function, bandwidthrequirements, signal to noise ratio, a method for one-dimensionalreconstruction, specific absorption rate, magneto-stimulation limits,and conditions for the construction of real instrumentation.

I. Fundamental Relations of MPI in One Dimension

A one-dimensional MPI system may be represented by a linear gradient −G[A/m/m] and a timechanging offset field H_(shift)(t) [A/m] such as whatwould be created by a Helmholtz coil (FIG. 14). If the gradient is zeroat the origin, the magnetic field at position u is as follows:

H(u,t)=H _(shift)(t)−Gu

Fundamentally, MPI relies on the rapid movement of a field free pointacross the sample to elicit a signal. The location of that field freepoint (FFP) can be represented by solving the above equation for H(x;t)=0. This yields:

x(t)=G ⁻¹ H _(shift)(t)

Substituting FFP position x(t), the magnetization at position u can berewritten:

H(u,x(t))=G(x(t)−u)

The magnetization M of a single magnetic nanoparticle in response to anapplied magnetic field in one dimension may be described by the Langevinequation:

M(H)=mρ

[kH]

where m [Am̂2] is the magnetic moment of the magnetic nanoparticle, k[m/A] is a property of the magnetic nanoparticle, H is the appliedmagnetic field, and ρ [particles/m³] is the nanoparticle density. Byusing the Langevin function, certain assumptions may be made. It may beassumed that the system responds whereby the total magnetic fieldinstantaneously aligns with the applied magnetic field (i.e., noparticle relaxation). It may also be assumed that there is noperturbation to the local magnetic field from nearby particles or E&Mproperties such as paramagnetism. Last, it may be assumed that theparticles obey the DC Langevin curve at RF and LF frequencies. Nowconsider an available magnetization density that is a continuousdistribution of magnetic nanoparticles, ρ(u,v,w). If is assumed that thenanoparticle distribution is distributed along the x axis and zeroelsewhere, then ρ(u)

(u,0,0). Thus, the magnetization at a single point u along thiscontinuous distribution when the FFP is at position x(t) is:

M(H)=M(u,x(t))=mρ(u)

[kG(x(t)−u)]

This one-dimensional magnetization can be converted into a flux, Φ, thatwould be detected by an inductive detector with −σ [T/A] sensitivity.Assuming the magnetization density only changes along u results in aconvolution:

$\begin{matrix}{{\Phi (t)} = {{- \sigma}\; m{\int{\int{\int{{\rho (u)}{\left\lbrack {k\; {G\left( {{x(t)} - u} \right)}} \right\rbrack}{u}{v}{w}}}}}}} \\{{= {{- \sigma}\; m\; {{\rho (u)} \otimes {\left\lbrack {k\; G\; u} \right\rbrack}}}}}_{u = {x{(t)}}}\end{matrix}$

where the u, v, and w axes correspond to the physical x, y, and z axes.The total magnetization in the system is a convolution of the magneticparticle density with a Langevin function kernel. This assumes that theMagnetic nanoparticles do not change the H field significantly. This isgenerally true since the average χ of magnetite at physiologicconcentrations is small. However, the point spread function of a 1D MPIsystem may not be the Langevin function. The MPI signal may be receivedusing an inductive detector that only sees changing magnetic fields.Visually described in FIG. 15, the MPI 1D signal equation may be derivedin volts, s(t):

s  ( t ) = -  Φ  t = σ   m   ρ  ( u ) ⊗ ′  [ k   G   u ] u = x  ( t )  k   G   x ′  ( t ) ( II  .1 )

For an image, a magnetic particle distribution convolved with a PointSpread Function (PSF) may be needed. The derivative of the Langevincurve may be assigned to be the system PSF. The extra terms may bedivided out, resulting in the MPI 1D image equation:

IMG  ( x  ( t ) ) = s  ( t ) σ   m   k   G   x ′  ( t ) = ρ ( u ) ⊗ ′  [ k   G   u ]  u = x  ( t ) ( II  .2 )

A simulated one-dimensional image that was reconstructed using thismethod is shown in FIG. 16. MPI is a Linear, Space Invariant (LSI)system.

II. The Langevin Equation

The Langevin function accurately describes the magnetization of anensemble of magnetic nanoparticles in response to an applied magneticfield. For N nanoparticles, each with magnetic moment m, themagnetization as a function of applied field, H, is:

$\begin{matrix}{{M(H)} = {N\; m\left( {k\; H} \right)}} \\{= {N\; {m\left( {{\coth \left( {k\; H} \right)} - \frac{1}{k\; H}} \right)}}}\end{matrix}$ where $k = \frac{\mu_{0}m}{k_{B}T}$

and kB is Boltzmann's constant, T is the temperature, μ₀ is the vacuumpermeability. The equation

$m = {M_{sat}\frac{\pi}{6}d^{3}}$

gives the magnetic moment of a spherical nanoparticle. For magnetiteSPIO nanoparticles, M_(sat)≈0.6 T/μ₀. The derivative of the Langevinfunction is a well behaved even function:

′  [ k   H ] = N   m  ( 1 ( k   H ) 2 - 1 sinh 2  ( k   H ) )

The kH product is a dimensionless value. In SI units, k has units of[m/A] with example values shown in the following table. The tableassumes round particles.

diameter (nm) k = μ₀m/k_(B)T (m/A) 15 0.26 × 10⁻³ 20 0.61 × 10⁻³ 25 1.19× 10⁻³ 30 2.05 × 10⁻³ 40 4.86 × 10⁻³ 50 9.49 × 10⁻³

For a spherical particle, k increases eight times when doubling theparticle diameter since k scales as the volume of the nanoparticle.Operating at reduced temperatures, such as 77K or 5K, may increase k andthe resulting resolution.

III. Spatial Resolution

From the image equation (Eq. II.2), the point spread function of the MPIprocess is the derivative of the Langevin function. Solving for the FullWidth at Half Maximum (FWHM) of this function, the intrinsic resolutionof the MPI process may be estimated analytically as FWHM_(L)≈4.16. Thisgives:

${FWHM}_{\infty} \approx {G^{- 1}\frac{4.16}{k}} \approx {\frac{4k_{B}T}{G\; m}\mspace{14mu}\lbrack m\rbrack}$

This implies that MPI resolution is proportional to the thermal energydivided by μ₀m. Resolution may be increased by changing the propertiesof the magnetic nanoparticles, k, or by increasing the gradient G. For aspherical particle,

$m = {\frac{\pi}{6}{\overset{\_}{M}}_{sat}d^{3}}$

where d is the particle diameter. The intrinsic resolution of the 1D MPIprocess may be calculated as a function of gradient strength andparticle size (FIG. 17 and Table I). Because of the cubic dependence oflinear spatial resolution with magnetic nanoparticle diameter, it may bepossible to use a large particle with resistive magnets. This may enableadjusting the intrinsic resolution of the system in real time bychanging the strength of the gradient.

TABLE I Resolution for given particle sizes and gradient strengths.Larger core diameters enable significantly improved resolution. A 5.1T/m magnetic field gradient is reasonable to build for small animals.particle size gradient strength (μ₀G) (nm) 1.3 T/m/μ₀ 2.6 T/m/μ₀ 5.1T/m/μ₀ 15   16 mm  7.9 mm  4.0 mm 20  6.6 mm  3.3 mm  1.7 mm 25  3.4 mm 1.7 mm 0.86 mm 30  2.0 mm  1.0 mm 0.50 mm 40 0.83 mm 0.41 mm 0.21 mm 500.42 mm 0.21 mm 0.11 mm

IV. Bandwidth

To design a MPI system, the bandwidth requirements to achieve thedesired resolution may be determined. The bandwidth required torepresent the Langevin function derivative may be approximated throughFourier analysis.

The derivative of the Langevin function is the PSF and defines theintrinsic resolution of the 1D MPI process. The derivative of theLangevin function may not have a simple Fourier transform, and so may beapproximated with a Lorentzian. To reasonable accuracy of <2% peakerror, the derivative of the Langevin function by a Lorentzian functionmay be approximated with FWHM ξ≈4.

${\zeta \left( {k\; H} \right)} = {\frac{2}{\pi}\frac{2}{4 + \left( {k\; H} \right)^{2}}}$

The maximum slew rate of a triangular, sinusoidal, or arbitrarilychanging FFP position may be modeled as a linearly changing FFPposition. A linearly ramping field with ramp rate R[m/s] gives a timevarying position x(t)=Rt. This corresponds to a time varying magneticfield of:

$\begin{matrix}{{H(t)} = {G\; {x(t)}}} \\{= {R\; G\; t}}\end{matrix}$

The magnetic field slew rate, RG, is the product of the scanning rateand the gradient size with SI units A/m/s. In certain cases, RG governsa number of important parameters including bandwidth requirement, SAR,and magnetostimulation.

For N particles located at the origin, ρ(x)=Nδ(x), the 1D signalequation II.1 and substituting a Lorentzian for the derivative of theLangevin curve gives:

s  ( t ) = σ   N   m   k   R   G  ′  [ k   R   G   t] ≈ σ   N   m   k   R   G   ζ  [ k   R   G   t ]

Taking the Fourier transform of s(t) yields:

${S(\omega)} \approx {\sigma \; N\; m\sqrt{\frac{2}{\pi}}{\exp \left( {- \frac{2{\omega }}{k\; R\; G}} \right)}}$

which describes the frequency content of the MPI experiment. ω≧0 sincethe MPI signal occurs at baseband and does not have a carrier frequency.

For a simpler representation of required bandwidth, the −3 dB bandwidthmay be solved for in Hertz:

$\begin{matrix}{F_{3\mspace{14mu} {dB}} = \frac{k\; R\; G\; {\ln (2)}}{4\pi}} \\{= \frac{M_{sat}d^{3}R\; G\; {\ln (2)}}{24}}\end{matrix}$

The MPI's bandwidth requirements increase linearly with nanoparticleproperties k, gradient strength G, and scanning rate R. Evaluating theFWHM for realistic parameters (R≈2400 [m/s], G=3:25=μ₀ [A/m/m], k=2×10⁻³[m/A]) that correspond to a particle with diameter d=30 nm, FOV=3 cm,and a sinusoidal excitation of f₀=25 kHz results in a F_(3dB)≈680 kHz.The relationship between the particle size, the intrinsic resolution,and the F_(3dB) bandwidth is shown in Table II. In some instances, theparticle determines the resolution and required bandwidth.

Since a real imaging system has a finite bandwidth, the received signalmay depend on the receive bandwidth of the receiver subsystem. Forexample, for receive bandwidth 4 f and a brick-wall filter, the receivedsignal in Fourier and real space is:

${S_{LPF}(\omega)} \approx {\sigma \; N\; m\sqrt{\frac{2}{\pi}}{\exp \left( {- \frac{2{\omega }}{k\; R\; G}} \right)}{{rect}\left( {\frac{1}{4\pi}\frac{\omega }{\Delta \; f}} \right)}}$s LPF  ( t ) ≈ σ   N   m   k   R   G  ′  [ k   R   G  t ] ⊗ 2  Δ   f   sin   c  ( 2  Δ   f   t )

This implies that limiting the bandwidth in frequency space with afilter may cause widening of the point spread function and ringing inthe real domain. With these bandwidth requirements, the receivebandwidth may be matched with the fundamental resolution of the system.For example, choosing a bandwidth whose FWHM is the same as theintrinsic resolution of the system may halve the achievable resolutionof the system when measured as the FWHM of a point source (FIG. 18( a)).From the graph, the intrinsic resolution is approached asymptotically.150% of the intrinsic resolution is not reached until the receiverbandwidth is Δf_(1:5)≈2.2 F_(3dB), and 110% of the intrinsic resolutionis not reached until Δf_(1:1)≈3.8 F_(3dB).

TABLE II Resolution and Bandwidth Scaling with respect to the particlesize. The values are normalized to a 30 nm particle. Smaller particleshave lower bandwidth requirements and poorer resolution. particle size(nm) resolution (mm) F_(3dB) (MHz) 15 7.9 0.13 20 3.4 0.30 25 1.7 0.5830 1.0 1.0 40 0.42 2.4 50 0.22 4.6

V. Signal to Noise Ratio

The maximum signal and noise may be calculated from first principles.From the MPI Signal Equation (Eq. II.1), the peak received signal for alinearly ramping magnetic field in volts is:

$s_{\max} = \frac{\sigma \; N\; m\; k\; R\; G}{3}$

The received noise may be from three sources, the noise figure of thepreamplifier, NF, the noise from the receive coil, R_(coil), and bodynoise, R_(body). Since the received signal is typically at a higherfrequency than 1/f noise present in semiconductor preamplifiers, thebody and coil noise may be modeled as spread across a noise bandwidth,Δf. Then, assuming Boltzmann noise, the standard deviation of theresistive voltage noise after the preamplifier is:

$(n) = {{N\; F\sqrt{4k_{B}\Delta \; {f\left( {{T_{coil}R_{coil}} + {T_{body}R_{body}}} \right)}}} \approx {N\; F\sqrt{4k_{B}\Delta \; f\; T_{coil}R_{coil}}}}$

which can be simplified by assuming coil noise dominance. If it isassumed that the bandwidth limitations increase the intrinsic FWHM ofthe imager by approximately half, the bandwidth requirements may beexpressed as a function of the 3 dB bandwidth: Δf=2F_(3dB)=kRGln(2)/(2π). Then, the Signal-to-Noise ratio (SNR), which was defined asthe peak signal divided by the standard deviation of the noise is:

${SNR}_{1D} = {\frac{s_{\max}}{(n)} = {\frac{\sigma \; N\; m\; k\; R\; G}{3N\; F\sqrt{4k_{B}\Delta \; {f\left( {T_{coil}R_{coil}} \right)}}} \approx {\frac{\sigma \; N\; m}{3N\; F\; k_{B}T_{coil}}\sqrt{\frac{{\pi\mu}_{0}m\; R\; G}{2{\ln (2)}R_{coil}}}}}}$

which assumes the nanoparticles are at the coil temperature, T_(coil).This describes the SNR for a single pass across the sample, and does nottake into account time averaging or acquisition time.A. 3D Signal to Noise Ratio with Averaging

The signal to noise ratio may be normalized to scan time to takeaveraging into account. If it is assumes a regular sampling of the fieldof view with readout in the x direction and acquisition of Ny and Nzlines in the y and z directions, respectively, during a total samplingtime of Ts, the number of averages may be estimated as:

$N_{avg} = \frac{R\; T_{S}}{{FOV}_{x}N_{y}N_{z}}$

Then, the three dimensional SNR can be estimated:

$\begin{matrix}{{SNR}_{3D} \approx {\frac{{\sigma\kappa}\; N\; m\; R}{3N\; F\; k_{B}T_{coil}}\sqrt{\frac{{\pi\mu}_{0}m\; G}{2{\ln (2)}T_{coil}}\frac{T_{s}}{{FOV}_{x}N_{y}N_{z}}}}} & \left( {{VI}{.1}} \right)\end{matrix}$

and as a function of particle diameter for a spherical particle:

$\begin{matrix}{{SNR}_{3D} \approx {\frac{{\sigma\kappa}\; N\; \pi^{2}d^{9/2}M_{sat}^{3/2}R}{36N\; F\; k_{B}T_{coil}}\sqrt{\frac{\mu_{0}G}{3{\ln (2)}R_{coil}}\frac{T_{s}}{{FOV}_{x}N_{y}N_{z}}}}} & \left( {{VI}{.2}} \right)\end{matrix}$

where κ≧1 arises from the increased SNR due to the contribution ofreceiver coils in multiple axes. The signal increases linearly with thescanning rate as faster scanning not only increases the raw signal, butalso increases the number of averages possible. The SNR increases withgreater time averaging, Ts, and decreases with increasing field of viewsize. This does not take into account the increased resistance of thebody as the frequency is increased since R_(body)∝ω², however at the lowfrequencies used, MPI is typically in a coil-noise dominated regime.

In some cases, increasing the particle diameter (Eq. VI.2) increases thesignal as SNR∝d^(9/2). This may also reduce the required gradientmagnitude required for the same resolution. In certain embodiments,increasing the saturation magnetization, M_(sat), of the magneticnanoparticles will also enhance SNR. The saturation magnetization may bechanged by changing the magnetic materials used.

System modifications can also increase the signal to noise ratio. Incertain embodiments, SNR increases with gradient strength, the FFPspeed, reducing coil noise, and improvement of the preamplifier noisefigure. In some instances, MPI's SNR increases as the scanning speed isincreased, even though the bandwidth requirements also increase. In somecases, reducing the gradient strength, G, also increases SNR for a fixedsample size because the scanning rate can be increased while maintainingthe same magnetic field slew rate.

From Equation VI.1, the SNR may be estimated for various scenarios inTable III. The calculations are made for specific applications with welldefined fields of view and resolutions at the SAR limit assuming asingle receive coil and sensitivity and noise values. For a smallanimal, the actual coil sensitivity value may be assumed to be σ=150uT/A and a noise of <n>=100 pV. For a human scanner σ=1.4 μT/A and<n>=1.8 pV. In certain embodiments, MPI should detect a single nanogramor less of magnetic material with reasonable SNR. These sensitivitynumbers are similar to those calculated in a simulation study on theoptimal SPIO core diameter for MPI imaging.

TABLE III Theoretical F_(3 dB), SAR and SNR, and detection limit forReal-Time (RT) and High Resolution (HR) scans for d = 30 nm particles atthe SAR limit of 4 W/kg. Note that we require at least BW = 2:2 F_(3 dB)to achieve 150% of the intrinsic resolution. Intrinsic G_(max) FOVResolution Time diameter 2.2F_(3 dB) SNR 1 Description R [m/s] [T/m/μ₀][cm] [mm] [s] [cm] [kHz] ng Fe heart (RT) 1150 1.3 20 2 × 4 × 4 1/5  34297 0.07 heart (HR) 577 2.6 20 1 × 2 × 2 30 34 297 0.3 brain (HR) 11152.6 18.4 1 × 2 × 2 60 18.4 574 1.0 extremity 1461 2.6 14 1 × 2 × 2 60 14752 1.9 (HR) mouse 6769 2.6 2.5 1 × 2 × 2 1/250 3 3482 1.9 (RT) mouse3451 5.1 2.5 0.5 × 1 × 1   60 3 3482 82 (HR)

VI. Sinusoidal Excitation

In certain embodiments, a MPI system excites the sample with atriangular excitation waveform. In some cases, since a triangularexcitation field would be composed of a sum of odd harmonics, a simplesinusoid that has limited frequency content may be used as theexcitation waveform.

In some cases, geometric isolation is performed. Geometric isolation mayreduce coupling between the transmit and receive coils by putting thereceive coil in a gradiometer configuration. This may also reducecoupled noise from outside the system bore. In some instances, aconductive sample in the bore induces eddy currents that induce a signalin the receive coil.

In certain embodiments, isolation between the transmit and receive coilsis performed using a combination of passive notch and high-pass filters.The combination may achieve million-fold or more reduction in theexcitation frequency from the received signal. In some instances, with asinusoidal excitation, the SNR decreases near the edges of the scanvolume because of the reduced magnetic field slew rate.

VII. Specific Absorption Rate

For a sinusoidally oscillating magnetic field, SAR in can beanalytically estimated for a cylinder as:

$P \approx \frac{\pi^{2}f_{1}^{2}B_{1}^{2}D^{2}}{8\rho \; s}$

where f₁ is the excitation frequency, B₁ is the excitation magnitude, Dis the diameter of the sample, ρ is the tissue resistivity at theexcitation frequency, and s is the specific gravity of the sample. If itis assumed sinusoidal excitation and a field of view, the excitationmagnitude and frequency may be estimated as:

$B_{1} = {\mu_{0}G\frac{FOV}{2}}$ and$f_{1} = {\frac{1}{2}\frac{R}{FOV}}$

which implies SAR in terms of scanning rate R and gradient strength Gis:

$P \approx \frac{{\pi^{2}\left( {\mu_{0}{RG}} \right)}^{2}D^{2}}{128\rho \; s}$

In some instances, the SAR presented in this manner does not depend onthe field of view.

At present, the United States Food and Drug Administration (FDA) limitsthe Specific Absorption Rate (SAR) deposition in the body to 4 W/kg. Themaximum magnetic field slew rate for a given sample size may be solvedfor. Since SNR increases with the magnetic field slew rate, thisrelationship may determine the optimal scanning rate. The maximum samplediameter may be a function of magnetic slew rate, as shown in FIG. 19.Large slew rates should be applied to the animal or patient at thelowest possible frequency that does not cause magneto-stimulation.

VIII. Slew Rate, and Magneto-Stimulation

A rapidly changing magnetic field in conductive tissue may induce anelectric field that can stimulate peripheral and cardiac nerves. Thefundamental law of magnetostimulation states:

${B_{1}(\tau)} = {B_{\min}\left( {1 + \frac{\tau}{\tau_{c}}} \right)}$

where B₁(f) is the peak amplitude for magnetostimulation as a functionof excitation time constant, τ, B_(min) is the minimal peak-to-peakexcitation amplitude for frequencies going to infinity, and τ_(c) is thechronaxie time constant. This model is similar to both FDA and IEC dB/dtregulations for Magnetic Resonance Imaging scanners.

Because current MPI scanners are constructed using a resonant excitationcoil, the minimum sinusoidal excitation frequency to preventmagnetostimulation of the peripheral nerves may be estimated. For asinusoidal excitation, the excitation rise time may be approximated as:

$\tau = \frac{1}{2\pi \; f_{1}}$

and so the fundamental law of magnetostimulation with a sinusoidalexcitation signal with respect to the desired magnetic field slew rateis:

$\begin{matrix}{\frac{B}{t} = {B_{\min}\left( {{2\pi \; f_{1}} + \frac{\tau}{\tau_{c}}} \right)}} \\{= {\mu_{0}{RG}}}\end{matrix}$

Solving for the optimal frequency of excitation for a magnetic fieldslew rate product while maximizing SNR:

$f_{1} = {\frac{\mu_{0}{RG}}{2\pi \; B_{\min}} - \frac{1}{2\pi \; \tau_{c}}}$

In certain embodiments, the system may operate with the maximum possibleslew rate that does not cause magnetostimulation. Because μ₀RG is largecompared to the slew rates used in MRI gradients, B_(min) is essentiallythe magnetic stimulation threshold. That is, B₁≈B_(min). For theperipheral nerves, the Z gradient may be estimated in an MRI withB_(min)≈12 mT for a peak-to-peak excitation magnitude of B_(pp)=2B_(min)and a chronaxie time constant of τ_(c)≈378 μs.

The optimal excitation frequencies for various sample sizes are shown inFIG. 19. This formulation may be important at lower excitationfrequencies as the ratio 1/(2πτ_(c)) 420 Hz. At the maximummagnetostimulation, excitation frequencies to achieve the SAR limitedmagnetic field slew rate, imaging a human heart can have up to 20 kHzline scan rate. In certain instances, a lower fundamental frequency iseasier to filter out because they take a smaller percentage of theavailable bandwidth. In some cases, imaging at the magnetostimulationlimit requires increasing the scanning frequency to prevent stimulationof the peripheral nerves in order to reach the desired magnetic fieldslew rates.

IX. Experiment 1

To test the principles described above, a zero dimensional MPIspectrometer was built as shown in FIG. 20. The system was constructedwith a excitation electromagnet and a biasing magnet. The outerelectromagnet generated the bias field, H_(bias), which simulatedplacing a point source sample at location u=G⁻¹H_(bias). Varying thebias field simulated moving the sample in a gradient field. The virtualFFP was scanned using a resonant transmit coil, H_(shift), and thesignal produced was received with a gradiometer receive coil. Using thesystem, the MPI signal was measured from an undiluted SPIO nanoparticlesample (Chemicell 50 nm fluidMAGD, Berlin, Germany). The point spreadfunctions were identical when diluted.

The bias coil was driven by a DC coupled audio amplifier (Crown M-600,Elkhart, Ind., USA) at field up to ±80 mT while dissipating 1 kW. Theresonant excitation coil generated 160 mT peak-to-peak at 6.23 kHz andwas driven by an audio amplifier (AE Techron LVC5050, Elkhart, Ind.,USA) with 5 kW of instantaneous power at a pulsed 1.5% duty cycle. Thesignal from the receive coil was digitized by a 16-bit data acquisitionsystem with a 1.25 MSPS sampling rate (National Instruments USB-6259,Austin, Tex., USA) controlled by custom software written in MATLAB(Mathworks MATLAB, Natick, Mass., USA).

X. Results

In FIG. 21( a), the received signal plotted as a function of FFPposition is shown. This is similar to the figure shown in FIG. 15. Themeasured data fit well to theoretical predictions. Normalizing to therelative speed of the FFP position showed that the point spread functiondid not change across the Field of View (FOV). The magnitude of themagnetic fields used here were beyond the magnetostimulation limit.Large excitation fields were used due to the small core diameters of theSPIO nanoparticles used in this experiment.

The measured point spread function was fit to what would be measured ifthere was a distribution of nanoparticle diameters. Assuming alog-normal distribution, the mean magnetic core diameter was estimatedto be 12.3 nm with a standard deviation of 3.2 nm. This correspondedwell to literature values for particles, which measured the corediameter from Resovist (Schering AG) at approximately 15.5 nm.

Multi-Dimensional X-Space Magnetic Particle Imaging A. MPI Signal Theory

In certain embodiments, the imaging system has Linearity and ShiftInvariance (LSI). In some instances, non-LSI systems may be analyzedusing standard signals and systems techniques such as convolution. Insome cases, imaging systems may use the temporal harmonic domain toanalyze the harmonics of the received MPI signal. Nonlinear magneticnanoparticles respond to a sinusoidal magnetic waveform with harmonicsignals at multiples of the excitation frequency. These harmonics aresuppressed sufficiently far from the center, or Field-Free-Point (FFP)of a field gradient, since the gradient field leaves the particles insaturation despite the RF excitation. Hence, the gradient field providesa method to localize harmonic response in 3D space. In certaininstances, the one-dimensional frequency-space signal can be describedusing Chebychev polynomials of the second kind convolved with themagnetization density. In certain embodiments, the Chebychev polynomialmodel is exact in one dimension, but extension to two and threedimensions may be an approximation.

A fundamental assumption of certain harmonic methods is that each pixelis interrogated over several cycles of the RF excitation. In some cases,this may not be accurate for faster scanning methods, where a singlepixel is scanned instantaneously only once. In certain embodiments, theimaging system does not require a repeating excitation, thus describingthe 1D MPI imaging process as an instantaneous scan through x-spacerather than a sinusoidal steady-state harmonic decomposition. In someinstances, the 1D x-space formalism can be applied to 2D and 3D.

B. Image Reconstruction in MPI

In certain embodiments, reconstruction techniques in MPI require apre-characterization of the magnetic nanoparticles whose signal responseis formulated into a system matrix. The system matrix may be comprisedof Fourier components of the temporal signal for every possible locationof a point source. E.g. for an image with N_(x)×N_(y)×N_(z) possiblepoints, the total number of elements in the system matrix will beN=N_(x)N_(y)N_(z)N_(c)N_(f) where N_(c) is the number of receive coilsand N_(f) is the number of Fourier components desired for thereconstruction. The system matrix can be measured physically using ananoparticle sample, or estimated using a model. However, in some cases,the system matrix is specific to the nanoparticle sample, andreconstruction may be depend on whether the nanoparticle behavesdifferently in tissue, if the system drifts, or if the model isinaccurate. In some instances, reconstruction may be achieved throughregularization and matrix inversion techniques such as singular valuedecomposition or algebraic reconstruction. In certain cases, thesolution is regularized to achieve high resolution while not amplifyingnoise when inverting the system matrix. In some instances, MPI imagereconstruction minimizes any loss of SNR.

In certain embodiments, MPI may be analyzed without a system matrixusing a narrowband MPI system that images multiple MPI harmonic mixingproducts and places them on a grid in real-space. In other embodiments,MPI may be analyzed using a theoretical formalism that may be validatedwith simulation and experiment. In some instances, a fast reconstructionalgorithm that computes the MPI image without matrix inversion andwithout a model based image reconstruction may be used. In theseinstances, the 1D image reconstruction method may be extended into 2Dand 3D.

I. Hypotheses for Multidimensional X-Space Magnetic Particle Imaging

In certain embodiments, reciprocity and linear shift invariant (LSI)imaging systems theory may be used to analyze the 1D MPI signal imagingprocess. In some instances, it may be assumed that the nanoparticlemagnetization instantaneously aligns with the applied local magneticfield. In certain cases, the MPI signal in one dimension is linear andspace invariant and can therefore be described as a convolution. In someinstances, the Point Spread Function (PSF) is the derivative of themagnetic nanoparticle's Langevin function. This analysis providedestimates for bandwidth requirements, which approach a megahertz fortypical imaging parameters. In certain embodiments, the maximum SNR willdepend on patient heating, and the maximum (partial) Field-of-View (FOV)will depend on magnetostimulation.

In certain embodiments, one-dimensional x-space MPI theory may beextended into two and three dimensions. In some instances, the magneticnanoparticles align instantaneously with the local magnetic field andthe loss of first harmonic information due to direct feedthroughcontamination is recoverable. In certain cases, for multidimensionalx-space, the linear 3D gradient field can be written as Gx where G is aninvertible matrix so that the gradient field uniquely identifies thelocation x in 3-space. In some cases, the real-world gradient field isinvertible, and 3D MPI is a linear and shift invariant imaging process.In certain instances, the analytic 3D point spread function of MPI maybe derived. In some cases, a fast image reconstruction algorithm thatrequires no calibration measurements or matrix inversion is used, suchthat the algorithm is both computationally efficient and robust tonoise. To apply the x-space formulation to real MPI systems, in somecases, the loss of the fundamental frequency breaks the strict LinearShift Invariance (LSI) properties. In certain instances, the lost firstharmonic information is fully recoverable using robust and noise-freeimage processing methods. The three hypotheses set forth above arejustified in detail below.

Hypothesis 1: The Instantaneous FFP is uniquely Defined in Space

MPI relies on a 3D linear gradient in the form:

$\begin{matrix}{{H(x)} = {Gx}} \\{= {\begin{bmatrix}{{- \alpha}\; G_{zz}} & G_{xy} & G_{xz} \\G_{xy} & {\left( {\alpha - 1} \right)G_{zz}} & G_{yz} \\G_{xz} & G_{yz} & G_{zz}\end{bmatrix}\begin{bmatrix}x \\y \\z\end{bmatrix}}}\end{matrix}$

where the vector x=[x y {dot of the image. In certain instances, asmooth and contiguous version of partial-FOV scans over the entire FOVmay be reconstructed using robust image processing methods with a smallamount of overlap in partial FOV scans. In certain cases, this allowsthe recovery of the lost baseline information without adding asignificant amount of noise.

II. Multidimensional Theory of MPI

For a continuous distribution of magnetic nanoparticles with densityρ(x)[particles/m³], from equations II.2 and II.3, the magnetizationdensity of ρ(x) nanoparticles located at position x when the FFP is atx_(s)(t) may written be as follows:

$\begin{matrix}{{M\left( {t,x} \right)} = {m\; {\rho(\; x)}{Z}k{{G\left( {x_{s},{(t) - x}} \right)}}\frac{G\left( {x_{s},{(t) - x}} \right)}{{G\left( {x_{s},{(t) - x}} \right)}}}} & \left( {{III}.\mspace{14mu} 1} \right)\end{matrix}$

and thus the total dipole moment is obtained by integrating themagnetization across the imaging volume is:

$\left. {{m(t)} = {\int{\int{\int{m\; \rho \; (u)Z{k}{G\left( {x_{s},\left( {t - u} \right)} \right.}}}}}} \right)\frac{G\left( {x_{s},{(t) - u}} \right)}{{G\left( {x_{s},{(t) - u}} \right)}}{u}$

This total dipole moment can be written as a spatial convolutioninterrogated at the instantaneous FFP location:

${m(t)} = {{m\; {{\rho(\; x)}**}*{\left\lbrack {k{{Gx}}} \right\rbrack}\frac{Gx}{{Gx}}}_{x = {x_{s}{(t)}}}}$

For an imaging system using an inductive detector, reciprocity can beused to calculate the received signal. For simplicity, orthogonalreceive coils can be assumed to be aligned with the x, y, and z axes ofthe instrument. Then, the sensitivity of the receive coils, −B₁(x)[T/A],would be a matrix of sensitivities. For the case for receive coils ineach of the x, y, and z-axes respectively, the sensitivity matrix wouldbe B₁(x)=[B_(1x)(x) B_(1y)(x) B_(1z)(x)]^(T). From reciprocity, thereceived signal vector is:

$\begin{matrix}{{s(t)} = {\frac{}{t}{\int{\int{\int{{B_{1}(u)}M\left\{ {t,u} \right){u}}}}}}} & \left( {{III}.\mspace{14mu} 2} \right)\end{matrix}$

To evaluate this derivative, the MPI signal equation may be evaluated.We begin by defining r and {circumflex over (r)}:

$\begin{matrix}{\begin{matrix}{r\overset{\Delta}{=}{k\; {G\left( {x,{(t) - x}} \right)}}} \\{\hat{r}\overset{\Delta}{=}\frac{r}{r}}\end{matrix}\begin{matrix}{{\overset{.}{r} = {k\; G_{x}^{+}}},(t)} \\{\overset{.}{\hat{r}}\overset{\Delta}{=}{\frac{\overset{.}{r}}{r} - {\frac{{\overset{.}{r}}^{\tau}r}{r}r}}}\end{matrix}} & \left( {{III}.\mspace{14mu} 3} \right)\end{matrix}$

{dot over (r)} can be decomposed into a tangential component, {dot over(r)}_(∥), and a normal component, {dot over (r)}_(⊥)=({dot over(r)}−{dot over (r)}_(∥)). Rewriting {dot over (r)} yields:

$\begin{matrix}{\hat{r} = {{\hat{r}}_{} + {\hat{r}}_{\bot}}} \\{= {{\left( {\hat{r} \cdot \frac{r}{r}} \right)\hat{r}} + \left( {\hat{r} - {\left( {\hat{r} \cdot \frac{r}{r}} \right)\hat{r}}} \right)}}\end{matrix}$

The derivative of the quasi-static Langevin function with vector-valued,time varying operand r:

  t   {  r  )  r ^ =  {  r  )  ( r . · r  r  2  r ) +  r  [ r . - r . · r  r  2  r ]

can be rewritten as a function of {dot over (r)}_(∥) and {dot over(r)}_(⊥):

$\begin{matrix}{\mspace{79mu} {{{\frac{}{t}\left\{ {r} \right)\text{?}} = {{\left\{ {r} \right)\text{?}} + {\frac{\left\{ {r} \right)}{r}{\overset{.}{r}}_{\bot}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & \left( {{III}{.4}} \right)\end{matrix}$

The derivative of the Langevin curve has two components, eachproportional to the tangential component or normal component of the FFPvelocity vector.

The equations above can be used to calculate the derivative of thesignal equation. From Eqs. III.1 and III.2 and definitions III.3, we canrewrite the MPI signal as:

$\left. {s\left\{ t \right.} \right) = {\frac{}{t}{\int{\int{\int{B_{1}\left\{ u \right)m\; \rho \; \left\{ u \right)\left\{ {r} \right)\hat{r}{u}}}}}}$

and evaluate the derivative using Eq. III.4:

 s  { t ) = ∫ ∫ ∫ ?  { u )  m   ρ   { u )  (  {  r  )  r . +  r   r . ⊥ )   u?indicates text missing or illegible when filed

Substituting for r gives us the convolution integral:

${s\left\{ t \right)} = {{\int{\int{\int{\text{?}\left\{ u \right)m\; \rho \; \left\{ u \right){\left\{ {r{{G\left\{ {{x_{s}\left\{ t \right)} - u} \right)}}} \right) \cdot \frac{\text{?}\left. {G\left\{ {{\text{?}\left\{ t \right)} - u} \right)} \right)}{{{G\left\{ {{\text{?}\left\{ t \right)} - u} \right)}}^{2}}}\left. {G\left\{ {{\text{?}\left\{ t \right)} - u} \right)} \right){u}}}}} + {\int{\int{\int{\text{?}\left\{ u \right)m\; \rho \; \left\{ u \right){\frac{\left\{ {k{{G\left\{ {{\text{?}\left\{ t \right)} - u} \right)}}\text{?}} \right.}{{k\left. {G\left\{ {{\text{?}\left\{ t \right)} - u} \right.} \right)}} \cdot {\quad{\left\lbrack {\text{?} - {\frac{\text{?}\left. {G\left\{ {{x_{s}\left\{ t \right)} - u} \right)} \right)}{\text{?}}\left. {G\left\{ {{\text{?}\left\{ t \right)} - u} \right)} \right)}} \right\rbrack {u}\text{?}\text{indicates text missing or illegible when filed}}}}}}}}}$

which yields:

$\begin{matrix}{\left. {s\left\{ t \right.} \right) = \left\{ {B_{1}{\text{?}**}*{\left( {k{{Gx}}} \right) \cdot \frac{\text{?}}{{{Gx}}^{2}}}{{Gx}}\text{?}\left\{ {B_{1}\left\{ x \right\} m\; \rho \; {\left\{ x \right\}**}*{\frac{\left\{ {k{{Gx}}} \right)}{\text{?}{{Gx}}} \cdot \left\lbrack {\text{?} - {\frac{\text{?}{{Gx}}}{{{Gx}}^{2}}\left. {Gx} \right)}} \right\rbrack}\text{?}\text{?}\text{indicates text missing or illegible when filed}} \right.} \right.} & \;\end{matrix}$

This form is not yet a simple PSF, and the FFP velocity magnitude ∥{dotover (x)}_(x)∥ and the FFP velocity unit vector

can be factored out by using an outer product vector identity(a·b)b=bb^(T)a. This yields the Generalized MPI signal equation:

$\begin{matrix}{\mspace{79mu} {{{s\left\{ t \right)} = {B_{1}\left\{ x \right)m\; \rho \; {\left\{ x \right)**}*k{\text{?}}h\left\{ x \right)\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & \left( {{III}{.5}} \right)\end{matrix}$

with Point Spread Function (PSF):

h  { x } =   k   Gx    Gx  Gx   ( Gx  Gx  ) 2  G + k  Gx   ( 1 - Gx  Gx   ( Gx  Gx  ) 2 )  G

The signal equation indicates the inductively received signal is thematrix multiplication of a matrix PSF with the FFP velocity vector,

The PSF does not change as a function of the FFP velocity magnitude,∥{dot over (x)}_(s)∥. If only the radially-symmetric scalar componentsof the PSF is considered, the PSF envelopes:

$\begin{matrix}{{ENV}_{T} = {\left( {k{{Gx}}} \right)}} & \left( {{III}{.6}} \right) \\{{ENV}_{N} = \frac{\left( {k{{Gx}}} \right)}{{kGx}}} & \left( {{III}{.7}} \right)\end{matrix}$

These envelopes seen in FIG. 23 give the maximum attainable resolutionin MPI. The higher resolution Tangential Envelope, ENV_(T), is thederivative of the Langevin equation. ENV_(T) defines the intrinsicresolution and bandwidth requirements for MPI. The lower resolutionenvelope, ENV_(N), is unique to generalized MPI and has a FWHM that is2.3× wider. The FWHM of both envelopes can be solved analytically as afunction of k or, alternatively, in terms of the particle diameter d:

$\begin{matrix}{{{FWHM}_{T} \approx {G^{- 1}\frac{4.16}{k}} \approx {\frac{25k_{B}T}{\mu_{0}G\; \pi \; M_{sat}d^{3}}\lbrack m\rbrack}}{{FWHM}_{N} \approx {G^{- 1}\frac{9.5}{k}} \approx {\frac{57k_{B}T}{\mu_{0}G\; \pi \; M_{sat}d^{3}}\lbrack m\rbrack}}} & \left( {{III}{.8}} \right)\end{matrix}$

In certain embodiments, ENV_(T) in equation III.6 may be derived intemporal frequency space, and may be derived in x-space. The secondenvelope in equation III.7 is unique to the generalized x-spaceformulation, and gives the resolution of the transverse component of thepoint spread function perpendicular to the FFP velocity vector.

The cubic relationship between resolution and particle diameter maydepend on the origin of MPI's signal. MPI's resolution relies on thenonlinear effect of a small applied magnetic field causing a SPIOnanoparticle tracer to magnetically saturate. The Langevin equationindicates that the field required to saturate a single magneticnanoparticle decreases with the nanoparticle's volume. As a result,resolution increases with the cube of the magnetic nanoparticlediameter.

A. MPI 3D Point Spread Function

The MPI process generates signals in multiple axes. In certaininstances, the inductive receiver coils are perpendicular to thephysical axes (x, y, and z) of the instrument, and the instrumentproduces images on an internal reference frame formed by vectorscollinear and transverse to the FFP velocity vector,

. The collinear and transverse images may be distinct from thetangential and normal components.

To understand the origin of the collinear and transverse components ofthe PSF, the vector components of h(x) may be examined. Supposing thatthe velocity vector is aligned with the x unit vector, i.e.

=ė₁. Then, the collinear component is:

h _(|)(x)=ê ₁ ·h(x)ê ₁

and the transverse components are:

h _(⊥,1)(x)=ê ₂ ·h(x)ê ₁

h _(⊥,2)(x)=ê ₃ ·h(x)ê ₁

where two perpendicular unit axes corresponding to the y and z axes, ê₂and ê₃, are arbitrarily chosen. The resulting components of the PSF areshown in FIG. 24. The collinear and transverse components of the PSFform an excellent basis set for image reconstruction.

If the FFP velocity vector is not oriented with one of the cardinaldirections of the instrument, the received images change with theorientation of the velocity vector

. To illustrate how the reference frame is oriented with an arbitraryFFP path, the collinear and transverse components of the PSF may beoriented with the velocity vector in FIG. 25.

While the equations presented herein are general, in certain instances,the point spread function in an algebraic equation may be obtained. Ifthe excitation vector ê₁ is fixed along the z-axis and a diagonalgradient matrix G=diag(G_(x),G_(y),G_(z)) is assumed, then the collinearPSF is as follows:

${h_{}\left( {x,y,z} \right)} = {{{\left\lbrack {{kH}\left( {x,y,z} \right)} \right\rbrack}\frac{G_{z}^{3}z^{2}}{{H\left( {x,y,z} \right)}^{2}}} + {\frac{\left\lbrack {{kH}\left( {x,y,z} \right)} \right\rbrack}{{kH}\left( {x,y,z} \right)}\left( {1 - \frac{G_{z}^{3}z^{2}}{{H\left( {x,y,z} \right)}^{2}}} \right)}}$

and one of the transverse PSFs on the receive axis aligned with thex-axis:

${h_{\bot{,1}}\left( {x,y,z} \right)} = {\left( {{\left\lbrack {{kH}\left( {x,y,z} \right)} \right\rbrack} - \frac{\left\lbrack {{kH}\left( {x,y,z} \right)} \right\rbrack}{{kH}\left( {x,y,z} \right)}} \right)\frac{G_{x}G_{z}^{2}{xz}}{{H\left( {x,y,z} \right)}^{2}}}$

where H(x,y,z)=√{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}. The PSFs for these equations are shownin FIG. 24.

The collinear component is similar to the real part of a Lorentzianfunction seen in NMR, and is an even function. The collinear componentis desirable and forms the bulk of the resolution and signal of the MPIimaging process. The collinear component is a vector sum of both thetangential and normal envelopes, ENV_(T) and ENV_(N), with the sharperenvelope aligned with the velocity vector.

The transverse component is similar to the dispersion or odd-valuedspectral component in NMR and is an odd function. The transversecomponent is a vector difference of the two point spread functionenvelopes. Across the diagonal (see FIG. 24), the signal received isprecisely

     PSF_(⊥) = ?({kH) − {kH)/kH).?indicates text missing or illegible when filed

As a result, the transverse component is significantly smaller inmagnitude than the collinear component.

III. X-Space Theory

In certain embodiments, x-space theory is used to predict the collinearPSF, intrinsic resolution, and bandwidth requirements. In some cases,Magnetic Particle Imaging images are based on a physical principle thatUltrasmall Superparamagnetic Iron Oxide (USPIOs) completely align andsaturate with the direction of any applied field greater than about 5mT. As a result, USPIOs can be selectively saturated by the applicationof a strong (6500 mT/m) magnetic field gradient. The magnetic field iszero at the center of the gradient at a location we term the Field FreePoint (FFP). Imaging occurs when the FFP is rapidly scanned across thesample, causing the magnetization of USPIOs passing through the FFP toflip and emit a signal that is detected using an inductive pickup coil.By gridding the signal to the instantaneous position of the FFP, animage of the nanoparticle distribution may be produced.

A. Point Spread Function

For a point source, i.e. a dirac delta source, the received image wouldbe that of the MPI point spread function. In certain cases, thecollinear component of the MPI signal is a PSF, h(x), multiplied by thevelocity vector, {dot over ({circumflex over (x)}[m/s], scaled by thevelocity of the FFP, ∥{dot over (X)}_(x)∥

$\begin{matrix}{\mspace{79mu} {{\text{?} = {B_{1}\left\{ x \right)m\; \rho \; {\left\{ x \right)**}*k{\text{?}}\text{?}\left\{ x \right)\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1)\end{matrix}$

where ρ(x) [particles/m³] is the magnetization density,

     m = ?M_(sat)d³[Am²]?indicates text missing or illegible when filed

is the magnetic moment of a single nanoparticle with saturationmagnetization M_(sat) [A/m] and diameter d [m], B₁ [T/A] is thesensitivity of the receive coils, and

     k = ? ?indicates text missing or illegible when filed

[m/A] is related to the properties of the magnetic nanoparticles. If theexcitation and reception vectors are fixed along the z-axis and assumean ideal diagonal gradient matrix G=diag(G_(x),G_(y),G_(z)), then we canwrite the collinear PSF

$\begin{matrix}{{{b_{1}\left( {x,y,z} \right)} = {{\text{?}\frac{\text{?}}{{H\left( {x,y,z} \right)}^{2}}} + {\frac{\left. {{kH}\left( {x,y,z} \right)} \right)}{{kH}\left( {x,y,z} \right)}\left( {1 - \frac{\text{?}}{{H\left( {x,y,z} \right)}^{2}}} \right)}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (2)\end{matrix}$

where H(x,y,z)=√{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}{square root over((G_(x)x)²+(G_(y)y)²+(G_(z)z)²)}. By dividing out the FFP velocitymagnitude, ∥{dot over ({circumflex over (x)}_(x)∥, and gridding thereceived signal, we obtain an image of the collinear component of thePSF

IMG ₁(x _(s)(f))1=s₁(t)/∥{dot over (X)}₁∥=ρ(x) * * *

·h ₁(x)Ż|_(x=x) ₁ _((t))

The collinear PSF is shown in FIG. 32( a). In our instrument, excitationand reception are collinear with the z-axis (See FIG. 31).

B. Resolution

To understand the intrinsic resolution of MPI, we must first defineresolution. Spatial resolution of the system is the ability toaccurately depict two distinct points of equal intensity in space. Thereare many classical criteria for resolution including Rayleigh, Schuster,Houston, and Buxton. Houston proposed a criterion where two points arejust resolved if two points are separated by the Full-Width at HalfMaximum (FWHM), which if the criterion used herein.

From the MPI PSF (Eq. 2), the PSF has two radially symmetric components,which may be termed the transverse and normal PSF envelopes

$\begin{matrix}{{ENV}_{T} = {\left( {k\sqrt{\left( {G_{x}x} \right)^{2} + \left( {G_{y}y} \right)^{2} + \left( {G_{z}z} \right)^{2}}} \right)}} & (4) \\{{ENV}_{N} = \frac{\left( {k\sqrt{\left( {G_{x}x} \right)^{2} + \left( {G_{y}y} \right)^{2} + \left( {G_{z}z} \right)^{2}}} \right)}{{k\sqrt{\left( {G_{x}x} \right)^{2} + \left( {G_{y}y} \right)^{2} + \left( {G_{z}z} \right)^{2}}}}} & (5)\end{matrix}$

From the Houston resolution criteria, the intrinsic resolution of theMPI process may be estimated analytically by solving for the FWHM ofthese envelopes for imaging the collinear component of the PSF ifexcitation occurs in the z-axis

$\begin{matrix}{{FWHM}_{T} \approx {G^{- 1}\frac{4.16}{k}} \approx {\frac{25k_{B}T}{\mu_{0}G_{z}\pi \; M_{sat}d^{3}}\lbrack m\rbrack}} & (6) \\{{FWHM}_{N} \approx {G^{- 1}\frac{9.5}{k}} \approx {\frac{57k_{B}T}{\mu_{0}G_{x,y}\pi \; M_{sat}d^{3}}\lbrack m\rbrack}} & (7)\end{matrix}$

That is, ENV_(T) gives the resolution transverse to the FFP velocityvector {dot over (X)}_(s), and ENV_(N) gives the resolution normal tothe FFP velocity vector (see FIG. 32( b)). This resolution implies thatresolution in MPI improves with the cube of magnetic nanoparticlediameter, and that significant gains in resolution could be achievedthrough new tracer agents specifically tuned for MPI. For the particlesshown herein, their apparent diameter is approximately 18±1.6 nm, whichcorresponds to transverse resolution of 1.6 mm resolution when imagedusing a 6.0 T/m gradient (see Table IV below).

TABLE IV Theoretical FWHM_(T) for particles of given diameter andstandard deviation in a 6 T/m gradient. Core Diameter [nm] FWHM [mm] 15± 3.4 2.2 17 ± 3.4 1.6 20 ± 3.4 1.1 25 ± 3.4 0.6 30 ± 3.4 0.3

C. Bandwidth

The highest resolution, and hence the highest bandwidth signal in MPI islimited by ENV_(T), which can be approximated by a Lorentzian function.A Lorentzian function has a well behaved Fourier transform, which can beused to estimate the bandwidth of the MPI signal. If the magnetic fieldslew rate is defined as RG [T/s/μ₀], then the signal spectrum is:

${S(\omega)} \approx {\sigma \; {Nm}\sqrt{\frac{2}{\pi}}{\exp \left( {- \frac{2{\omega }}{kRG}} \right)}}$

and solving for the half power point of the signal bandwidth gives:

$\begin{matrix}{F_{3\; {dB}} = \frac{{kRG}\; {\ln (2)}}{4\pi}} \\{= \frac{M_{sat}d^{3}{RG}\; {\ln (2)}}{24}}\end{matrix}$

Because SNR decreases with increasing bandwidth, the minimum bandwidthnecessary may be used. However, the reception may not be limited to onlythe 3 dB bandwidth of the signal, as this may result in spatialblurring. Assuming a brick-wall filter the bandwidth required to widenthe FWHM by only 10% may require a receive bandwidth of BW_(recv)≈3.8F_(3dB).

D. Linearity/Shift Invariance

The adiabatic model assumed here is theoretically required for shiftinvariance. The adiabatic model assumes instantaneous rotation of themagnetic moments in response to an applied magnetic field. In someinstances, the rotation of a magnetic nanoparticle lags the appliedfield due to both Neel and Brownian relaxation. This lag can bedescribed by a relaxation time constant, which is in the range of 1-30μs for the Brownian relaxation dominated nanoparticles used in MPI. Asshown in the experimental results, the adiabatic model still producesimages with high resolution and SNR when using Resovist.

MPI interrogates magnetic nanoparticles with a rapidly varying magneticfield. This applied field can induce a signal in the receive coil manyorders of magnitude larger than the nanoparticle signal. This appliedfield may be a sinusoid, whose frequency may be termed as the“fundamental frequency”. To avoid overwhelming the small nanoparticlesignal, the fundamental frequency may be filtered out using band-stopand high-pass filters. In some instances, this breaks the LSI propertiesof the system because information used for reconstruction has beenremoved.

Experimental

From a basis in x-space theory, an imaging technique was developed thatenabled the acquisition of an intrinsic MPI image. The technique mayinclude three components: (1) imaging pulse sequence, (2) gridding, and(3) image assembly. This technique resulted in a LSI system, and theresulting image was the magnetization density convolved with the PSF.

A. Pulse Sequence

The path of the FFP through x-space may be considered as a MPI pulsesequence. There are various pulse sequences possible in MPI, includingbut not limited to raster, Lissajous, polar-rose, etc., and the path ofthe FFP may affect the SNR of the resulting image.

A raster pulse sequence was used (see FIGS. 33( a) and 33(b)) for thisx-space scanner as it was straightforward to implement with the imageequation (Eq.3). A raster scan simplified reconstruction because the FFPvelocity unit vector

can be considered a constant since rapid FFP movement was limited to thez-axis. Slow movement in x, y and z was done mechanically.

MPI systems can use a large gradient field to increase resolution at theexpense of reducing the field of view of a scan. For example, thescanner described here generated a 30 mT_(peak-peak) excitationamplitude on top of a 6.0 T/m gradient, which moved the FFP by 5 mm.This excitation amplitude exceeded the limits of magnetostimulation fora chest scanner, but would not exceed the limits of magnetostimulationin an extremity or head scanner.

In order to scan a larger FOV, the system acquired partial FOVs thatwere later assembled into a full image. A partial FOV was acquired byrapidly scanning 5 mm in z while simultaneously mechanically translatingthe sample 1.5 cm in y. To acquire the full image, the sample wasstepped down the bore in z so that the partial FOVs overlapped. It waspossible to move the FFP electronically through the addition of highamplitude slow field shifting magnets, which were previously implementedin a mouse scale narrowband MPI system.

B. Gridding

A systematic method for converting the signal returned by the pulsesequence into x-space was developed. To do this, the signal wasprocessed to ensure the phase linearity, and then the received signalwas gridded to the instantaneous location of the FFP.

The magnetic signal generated by the nanoparticles was received by aninductive receive coil. The received signal was contaminated by a largeinterfering sinusoid at the fundamental frequency. The fundamentalfrequency was removed by a notch filter and the remaining signal waspre-amplified. The signal was conditioned by a High-Pass Filter (HPF)followed by a second stage of amplification.

In certain instances, the receive subsystem did scramble the phase ofthe received signal, as phase corresponds to location in x-space. Ahigh-Q notch filter built as shown in FIG. 35 was tuned to minimallychange the phase of the signal. The addition of a HPF improved noiserejection by removing low-frequency interferers including mains noise,signal from slow FFP movement magnets, remaining fundamental signal, and1/f noise. Although, the HPF broke the phase of the received signal, itwas possible to reverse this phase accrual through digital anti-causalfiltering or a phase-matched analog or digital all-pass filter. Since aButterworth filter was simple to design in the digital domain, anon-causal filtering step was used to recover the phase. The phasecorrected signal s_(ph) can be expressed as

s _(ph)(t*)=HPF[s(t*)]

where t*=−t to reverse time. In certain embodiments, the high passfilter was an eighth order Butterworth with a 25 kHz cutoff frequency.

Using a simple method of gridding, the phase corrected signal wastransformed from the time domain to the image domain. This method didnot require regularization, optimization techniques, or prior knowledgeof the magnetic response of the tracer. Following the scan, the phasecorrected signal was divided by the instantaneous FFP velocity, and theamplitude corrected signal was gridded to the instantaneous location ofthe FFP. That is

$\mspace{79mu} {{{IMG}\text{?}} = \frac{\text{?}}{\text{?}}}$?indicates text missing or illegible when filed

IMG(x) was interpolated and averaged using a nearest neighbor algorithm.

C. Image Assembly

When imaging with a partial FOV, filtering the received signal removesthe LSI properties of the system. In certain instances, the HP F of thetime domain signal may be considered as a loss of low spatial frequencyinformation to account for the loss of low frequencies. For the loss oftemporal frequencies near the fundamental frequency, this spatial signalloss may be approximated as a DC offset so that the mean signal valuewas zero. In certain embodiments, this means that if multipleoverlapping partial FOVs were acquired, the overlap between signals maybe found that minimizes their overlap error. Since only a constant DCoffset was lost and assuming boundary conditions at the endpoints of thescan, the DC offsets necessary to optimally overlap the partial FOVs maybe estimated. Adding the overlapped partial FOVs weighted by SNR of eachpoint returned an assembled image.

Relationship Between X-Space Theory and Bandwidth, Achievable Resolutionand SNR

X-space MPI theory indicates that various parameters may depend on thereceive bandwidth. In some instances, a wider bandwidth will improveresolution. In certain cases, continuing to increase the bandwidthbeyond what is necessary given a scan rate and magnetic field strengthwill not continue to improve the achievable resolution. In someembodiments, the bandwidth may be linked to the SNR of the system.Continuing to increase the bandwidth may decrease SNR in certainembodiments. In some cases, reducing system bandwidth may improve SNR,irrespective of whether the system is coil noise or preamplifier noisedominated.

A. X-space MPI Theory

The x-space formalism for MPI gives a signals and systems framework tounderstanding the raw MPI signal, and how to convert the time domainsignal into the spatial domain without matrix inversion. X-space MPI maybe used to derive system resolution, bandwidth, and SNR from firstprinciples. In certain cases, the temporal harmonic domain may be used,where the harmonics of the received MPI signal in response to asinusoidal excitation frequency. X-space analysis may differ from theseanalyses by describing the MPI signal in the time domain withoutanalysis of harmonics.

From the assumption that the magnetic nanoparticles respondadiabatiacally to the applied magnetic field, in one dimension, the MPIsignal in volts can be written as a convolution

s(t)=B ₁ mρ(x)*

(kGγ)∥_(x=x) _(s) _((t)) kG{dot over (γ)}x(t)  (3.1)

where B₁ [T/A] is the sensitivity of the receive coil, m [Am²] is themagnetic moment of a single nanoparticle, ρ [particles/m³] is thenanoparticle density,

is the derivative of the Langevin function, k [m/A] is related to thenanoparticle properties, G [A/m/m] is the magnetic field gradient, x isthe location in real space, and x_(s)(t) is the location of the FFP.

The rate of change of a triangular, sinusoidal, or arbitrarily changingFFP position may be modeled as a linearly changing FFP position. Alinearly ramping field with ramp rate R[m/s] gives a time varyingposition x_(s)(t)=Rt. This corresponds to a time varying magnetic fieldof

$\begin{matrix}{{H(t)} = {{Gx}_{s}(t)}} \\{= {RGt}}\end{matrix}$

where we term RG [A/m/s] the magnetic field slew rate.

For N particles located at the origin, ρ(x,y,z)=Nδ(x)δ(y)δ(z), the 1Dsignal equation 3.1 and approximating the Langevin curve as a Lorentziangives

     s(t) = B₁NmkRG[kRGt] ≈ B₁NmkRG ζ(kRGt)      where$\mspace{79mu} {{{\left. {kH} \right\rbrack} \approx {\text{?}\left. {kH} \right\rbrack}} = {\frac{2}{\pi}\frac{2}{4 + \left\{ {kH} \right)^{2}}}}$?indicates text missing or illegible when filed

Taking the Fourier transform of s(t) yields an estimate of MPIbandwidth, which is a smooth spectrum (see FIG. 45)

${S(\omega)} \approx {B_{1}{Nm}\sqrt{\frac{2}{\pi}}{\exp \left( {- \frac{2{\omega }}{kRG}} \right)}}$

The harmonics typically seen in the received spectrum are an artifact ofthe repeating excitation, and not fundamental to MPI (see FIG. 46). Thisis because the average position of the FFP moves slowly when compared tothe rapid movement of the FFP in a typical scanning sequence. In onedimension, this may be similar to sampling the field of view repeatedlyin time. This is similar to convolving the signal for a single passacross the FOV with a repeating Shah function. However, the Fouriertransform of the Shah function samples the received signal spectrum atmultiples of the repetition frequency, leading to harmonics in thereceived spectrum. If the FFP was moved rapidly so that the same fieldof view is not repeatedly sampled, the received spectrum would besmooth.

With a suitable mapping of the time domain to the spatial domain usingx-space theory, the Fourier transform of the spatial domain is in thespatial frequency domain. For the case of the linear ramp excitation,the spectra of the temporal and spatial frequency domains are identical.This enables direct measurement of the Modulation Transfer Function(MTF), which enables comparison of MPI resolution with other imagingmodalities such as x-ray and magnetic resonance imaging.

B. Bandwidth and Resolution

The finite receiver bandwidth of a real system reduces the bandwidthfrom the intrinsic resolution to what may be termed the achievableresolution. To see this mathematically, a receive bandwidth, Δf, with abrick-wall filter may be assumed. Then, the received signal in Fourierand real space is:

${S_{LPF}(\omega)} \approx {B_{1}{Nm}\sqrt{\frac{2}{\pi}}{\exp \left( {- \frac{2{\omega }}{kRG}} \right)}{{rect}\left( {\frac{1}{4\pi}\frac{\omega }{\Delta \; f}} \right)}}$s_(LPF)(t) ≈ B₁NmkRG[kRGt] * 2Δ fs inc(2Δ ft)

This implies that limiting the bandwidth in frequency space with alowpass filter will cause widening of the PSF, reducing the achievableresolution. The intrinsic resolution is approached asymptotically as thereceive bandwidth is increased. 150% of the intrinsic resolution is notreached until the receiver bandwidth is Δf_(1.5)≈2.2F_(3dB), and 110% ofthe intrinsic resolution is not reached until Δf_(1.1)≈3.8F_(3dB). If aGaussian low pass filter was assumed instead, the profile may beapproximated as a Voigt profile, which is a Lorentzian convolved with aGaussian, for which there may be closed form approximations.

C. Bandwidth and SNR

The signal generated by the magnetic nanoparticles is detected andamplified in the receive chain. The receive chain has two primary noiseproducing components, the pre-amplifier and the inductive receiver coil.In certain embodiments, it may be desirable to minimize the noise addedby these two components, which are dominated by the voltage noise of thepreamplifier by the resistance of the receive coil. When designing thereceive chain the pre-amplifier and receive coil design may depend onthe receive bandwidth.

In certain embodiments, MPI preamplifiers are not matched to a resistiveor resonant impedance. A pre-amplifier suitable for MPI may have lownoise amplification across a wide bandwidth under 1 MHz. For example, acommercially available ultra-low noise FET preamplifier that has founduse is the SA-220F5 (NF Corporation, Yokohama, Japan), which has aninput referred voltage noise of e_(n)=0.5 nV/√{square root over (Hz)},current noise of i_(n)=200 fA/√{square root over (Hz)}, and an inputcapacitance of 65 pF. If a receive coil with a relatively largeinductance of 500 pH was used, the current noise would correspond to anequivalent voltage noise of 0.1 nV/√{square root over (Hx)} at 150 kHz.At the low frequencies used in MPI, even for a high impedance receivecoil, the total input referred voltage noise of a FET input stage isdominated by the voltage noise rather than the current noise. Since theMPI received signal is typically at a higher frequency than 1/f noisepresent in Si-BJT and Si-JFET preamplifiers, the body and coil noise maybe modeled as spread across a noise bandwidth, Δf. The noise of aresistive receive coil may be modeled as a broadband Johnson noise.Assuming that the AC resistance remains flat and coil noise dominance,the input referred resistive voltage noise can be calculated across thenoise bandwidth

${\langle n\rangle} = {\sqrt{\left( {\text{?} + {4{k_{B}\left( {{T_{coil}R_{coil}} + {T_{b}R_{b}}} \right)}} + {i_{n}^{2}Z_{coil}}} \right)\Delta \; f} \approx \sqrt{\left( {\text{?} + {4k_{B}T_{coil}R_{coil}}} \right)\Delta \; f}}$?indicates text missing or illegible when filed

where e_(n) [V/√{square root over (Hz)}] is the voltage and currentnoise amplitude per unit bandwidth of the preamplifier, k_(B) isBoltzmann's constant, T_(coil) [K] is the temperature of the coil, andR_(coil) [Ω] is the resistance of the coil, and T_(b) is the bodytemperature, and Rb is the body resistance. The noise current i_(n)[A/√{square root over (Hz)}] may be ignored because noise currents tendto be negligible at the low frequencies used in broadband MPI.

Since the dominant noise sources in the front end electronics have beenestimated above, the Signal to Noise Ratio (SNR) may be calculated asfollows:

$\mspace{79mu} {{SNR} \approx \frac{\text{?}}{\sqrt{\left( {\text{?} + {4k_{B}T_{coil}R_{coil}}} \right)\Delta \; f}}}$?indicates text missing or illegible when filed

In certain embodiments, it may be desirable to achieve coil noisedominance where the Johnson noise from the coil resistance is greaterthan the amplifier noise contribution, i.e. 4K_(B)T_(coil)R_(coil)>e_(n)², as the body noise contribution is small under 1 MHz. In certainembodiments, cooled copper and superconducting receive coils mayfacilitate achieving coil noise dominance.

Gridding and Fundamental Frequency Recovery

In certain embodiments, the method of imaging magnetic particles inx-space

MPI includes gridding and fundamental recovery. Gridding may be used totransform the received signal from the time domain to the image domain.The technique of a partial FOV can be used to estimate the low frequencycontent that is lost when the fundamental frequency is filtered out.These x-space reconstruction techniques do not require regularization,optimization techniques, or prior knowledge of the magnetic response ofthe tracer.

A. Reconstruction Methods: Gridding

Gridding in MPI is the process of sampling the received signal s(t) ontoa grid in real space, or x-space, which corresponds to the instantaneouslocation of the FFP. In some cases, the collinear and transversecomponents are gridded separately. To do this, the image may beseparated into collinear and transverse signals:

s _(∥)(t)=

·s(t)

s _(⊥,1)(t)=(

×ê₁)·s(t)

s _(⊥,2)(t)=((

×ê₁)×ê ₂)·s(t)

An arbitrary unit vector ê₁ may be chosen to cross with the velocityunit vector

to build a perpendicular basis set of transverse vectors. Choice of thisarbitrary vector assumes that ê₁ and

are not collinear, i.e.

×ê₁≠0.

If the pulse sequence is designed so that the velocity unit vector

is constant, e.g. with fast movement only in one direction, gridding ofthe collinear and transverse signals may be simplified. Ignoring thereceiver coil sensitivity and choosing arbitrary unit vectors ê₁ and ê₂not collinear with the FFP velocity

gives the Generalized MPI Image Equation:

$\begin{matrix}\begin{matrix}{{{IMG}_{}\left( {x_{s}(t)} \right)} = {{s_{}(t)}/{{\overset{.}{x}}_{s}}}} \\{= {{{{\rho (x)}**}*{{\overset{\hat{.}}{x}}_{s} \cdot {h(x)}}{\hat{\overset{.}{x}}}_{s}}_{x = {x_{s}{(t)}}}}}\end{matrix} & \left( {{IV}{.1}} \right)\end{matrix}$

where the magnitude of the FFP velocity is normalized. Similar griddingcan be done for the remaining transverse images. This image equation issimilar to the k-space analysis of MRI, but here the scanning occurs inx-space rather than in k-space, so no Fourier Transform is required.

B. Reconstruction Methods: Fundamental Recovery

MPI interrogates magnetic nanoparticles by subjecting the sample to arapidly varying magnetic field. In certain embodiments, the appliedmagnetic field contaminates the received signal as the applied fieldinduces a signal in the receive coil that is many orders of magnitudelarger than the signal generated by the magnetic nanoparticles. Thisapplied field is typically a sinusoid, whose frequency we term the“fundamental frequency”.

To correct for the loss of the fundamental frequency, a partial FOVtechnique may be used. MPI systems can use a large gradient field toincrease resolution at the expense of reducing the FOV of a scan. Forexample, the scanner described herein generated a 30 mT_(peak-peak)excitation amplitude on top of a 6 T/m gradient, giving a total FOV ofabout half a centimeter. In certain embodiments, this excitationamplitude may exceed the limits of magnetostimulation for a chestscanner and may be near the magnetostimulation limit for an extremityscanner. However, partial FOV images can be taken that can be combinedtogether for the full FOV by slowly moving the average position of theFFP mechanically or with an electromagnet.

When scanning an image with overlapping partial FOVs, it may be possibleto recover the lost fundamental signal, which is important to shiftinvariance. For instance, a plurality of partial FOV images may becombined into a larger field of view image. In certain embodiments,high-pass filtering of the time-domain signal as a loss of low-spatialfrequency information may be used. For the loss of temporal frequenciesnear the fundamental frequency, this spatial-signal loss can beapproximated as a DC offset. In some instances, if multiple overlappingpartial FOVs are acquired, the overlap between signals may be found thatminimizes their overlap error. Since only a constant DC offset was lostand assuming boundary conditions at the endpoints of the scan, theresulting reassembled image will approximate the original spatialconvolution.

Experiment 2 C. Experimental Methods

A three-dimensional MPI scanner was built, as shown in FIGS. 26( a) and26(b). The system was constructed with a permanent magnet gradient (6T/m down the bore and 3 T/m transverse to the bore) and an excitationcoil in one dimension collinear to the bore. The FFP was rapidly scannedusing the resonant transmit coil and the signal produced was receivedwith a receive coil wound collinear to the transmit coil. The receivecoil received the collinear component of the vector PSF. The transmitand receive coils were collinear with the larger gradient along thebore, which was twice the magnitude of the gradient transverse to thebore. The collinearity of the coils was chosen for ease of constructionbut resulted in an intrinsic resolution in the transverse direction thatwas approximately four times less than in the collinear direction (seeEq. III.8).

The resonant excitation coil generated 30 mT peak-to-peak at 20 kHz andwas driven by an audio amplifier (AE Techron LVC5050, Elkhart, Ind.,USA) with ˜5 kW of instantaneous power at a pulsed 2% duty cycle. Thesignal from the receive coil was filtered by a passive notch filter,amplified by a battery powered preamplifier (SR560, Stanford ResearchSystems), and high-pass filtered at 35 kHz (SIM965, Stanford ResearchSystems). Following the analog signal chain, the signal was digitized bya 16-bit data acquisition system with a 1.25 MSPS sampling rate(National Instruments USB-6259, Austin, Tex., USA), phase corrected, andlow-pass filtered at 400 kHz. The system was controlled by customsoftware written in MATLAB (Mathworks MATLAB, Natick, Mass., USA).

The 30 mT peak-to-peak excitation enabled a partial FOV of approximately0.5 cm along the z-axis. The signal for the received partial FOV wasgridded to the instantaneous location of the FFP and assigned to aphysical location on the phantom. The phantom was stepped in 1 mm stepsalong the z-axis for 4 cm, which acquired a partial FOV line scan ateach step. The line scans were reassembled by estimating the missingfundamental to generate an assembled full FOV of 4.5 cm along thez-axis. A total of 20 line scans were taken by moving in 1 mm stepstransverse to the bore, for a full FOV of 2 cm in the y-axis. Phantomswere constructed using 400 micron ID tubing filled with undiluted SPIOtracer (Resovist, Bayer-Schering).

D. Results & Discussion

In FIGS. 27( a) and 27(b) experimental data is shown, showing theone-dimensional scan of a point source before and after fundamentalrecovery. The fundamental was recovered by estimating the DC offset ofeach segment so that a maximally smooth image was produced.

In FIGS. 28( a), 28(b), 30(a) and 30(b), images measured with thex-space MPI imager are shown. As seen in the images of the PSF in FIGS.28( a) and 28(b) and line scans in FIGS. 29( a) and 29(b), the FWHM inthe normal axis was 4.6 times wider than the FWHM along the axis of theimager. The lower resolution in the normal axis agreed with thetheoretical prediction in Eq. III.8. The measured FWHM being wider thanthe theoretical prediction may be attributed to the nanoparticlebehaving differently than in our model, and the phantom being a linesource rather than a point source. In certain embodiments, the imagingsystem may orient the magnetic fields differently so as to optimize theshape of the PSF. The “CAL” phantom image shown in FIG. 30( b) was anative two-dimensional MPI image without any sharpening or deconvolutionand with full recovery of the fundamental frequency, which maintainedthe LSI properties of the system. As seen in FIG. 30( b), the receivedimage for the “CAL” phantom approximated the phantom itself.

The x-space technique described herein is a MPI imaging process. Threehypotheses were used, that the gradient creates a single FFP, theadiabatic Langevin model, and that the loss of the low frequencies isrecoverable. These three hypotheses allowed for the analysis of the MPIimaging process. The experimental evidence presented herein showed thatloss of low frequency information was recoverable.

In certain embodiments, x-space theory did not require a repeatingsinusoidal excitation or specific Lissajous pulse sequence. The x-spaceformulation also included an image reconstruction technique that wasrobust, scalable, and faster than inversion of the system matrix and itdid not require pre-characterization of the magnetic nanoparticles orsystem. In certain instances, the computation required for x-spacereconstruction, required only a scaling and gridding. For example, thereconstruction code may reconstruct the received signal faster than ananalog-to-digital converter was able to digitize data.

The intrinsic FWHM resolution predicted by the x-space analysis agreedwith the non-deconvolved resolution limit. The 2D and 3D analysispresented herein extend these initial analyses to show that theintrinsic resolution changes with the orientation of the FFP movementsequence. In certain embodiments, increasing the apparent diameter ofthe SPIO magnetic core to 25 or 30 nm and compensating for relaxationeffects increased the intrinsic resolution of the image.

In certain embodiments, MPI is a LSI system with the three hypothesesdiscussed herein, and the experimental results showed that the recoveryof the first harmonic enabled experimental MPI systems to be accuratelymodeled as LSI.

Experiment 3

Experiments were performed to construct and test a MPI device based onx-space theory as described herein.

Imaging Hardware

The design criteria and the construction of a small scale MPI device totest x-space theory are described herein.

A. Gradient

MPI requires a large magnetic field gradient that selectively saturatesmagnetic nanoparticles. Since resolution improves strongly withincreasing nanoparticle core diameter and weakly with increasingmagnetic field gradient strength, each nanoparticle size requires adifferent strength gradient to achieve a target resolution. In certaininstances, using larger nanoparticles may increase resolution. In someinstances, superparamagnetic nanoparticles may have an average diameterof 20 nm. In some cases, a commercially available nanoparticle tracer,Resovist (Bayer-Schering) was used, which has a signal very similar to ananoparticle with 17±3.4 nm core diameter.

A 6.0 T/m (3.0 T/m transvere to the bore) gradient was built withpermanent magnets using the magnet configuration shown in FIG. 31. Thegradient was built using two opposed NdFeB ring magnets (ID=8.89 cm,OD=14.6 cm, THK=3.2 cm) mounted on G10 backing plates. The permanentmagnets generated a 6.0 T/m gradient down the bore, and 3.0 T/mtransverse to the bore. With Resovist particles, this enabled 1.6 mmresolution down the bore (see Eq.6). Because of the reduced transversefield gradient strength and since excitation was not performed in atransverse axis, the resolution equation (Eq. 7) predicted that thetransverse resolution was 7.4 mm, or 460% greater than the collinearPSF. Inside the bore was a 2.5 mm thick, water cooled, copper eddycurrent shield that gave both mechanical rigidity and magneticallyisolated the bore from the surrounding environment.

In some cases, the temperature coefficient of rare Earth magnets wastypically 1 ppt/C, which would require milliKelvin temperature stabilityfor NMR/MRI applications. However, for MPI applications, several degreetemperature variations were well tolerated.

B. Excitation and Reception

In certain embodiments, the transmit chain excited the sample with apure sinusoid with no energy content above the excitation frequency. Thereceive chain received a wide bandwidth signal, and at the same timesuppressed the fundamental frequency. The transmit-receive filters weredesigned as shown in FIG. 35.

The resonant transmit coil (f₀=19 kHz) was wound with 10 gauge squaremagnet wire and was driven by a high power linear amplifier filtered bya three stage low pass filter. The transmit filter achieved 60 dB ofisolation at 2× the fundamental signal and 65 dB isolation at 3× thefundamental signal. The transmit coil generated 30 mTpp with a peakpower output of 5 kW. In some instances, a portion of the power outputof the transmit coil was dissipated as heat in the water cooled eddycurrent shield. The receive coil was wound in a gradiometer-likeconfiguration inside the transmit coil to minimize total shared flux.The receive coil had a sensitivity of B₁=XX T/A to the sample inside thebore and a DC resistance of XX ohms. The signal from the receive coilwas notch filtered by a resonant filter, and amplified by abattery-powered low noise pre-amplifier (Stanford Research SystemsSRS560). The signal was further conditioned by a noise matched 8th orderanalog Butterworth high-pass filter (F_(3dB)=25 kHz, Stanford ResearchSystems SIM965), followed by a second stage of amplification (StanfordResearch System SIM911). The signal was digitized at 1.25 MSPS (NationalInstrument, NI-6259), digitally phase corrected, low pass filtered at200 kHz, and gridded to the instantaneous position of the FFP.

In certain cases, it was desirable to have a high power transmit filterwith sufficient rejection of noise above the fundamental frequency. Forthe phantoms shown in the accompanying figures, the feedthrough of thesecond and third multiples of the fundamental frequency was about halfof the size of the MPI signal from the phantoms. To counteract this, abaseline image with no sample in the bore was taken and subtracted fromthe received signal. However, in some instances, the feedthrough driftedand remained the dominant noise source.

Results and Discussion

The signal received by the x-space scanner had high SNR andrepeatability. In FIGS. 36( a) and 36(b), the phase corrected signalreceived by the x-space imager for a point source sample located at theorigin is shown. The amplitude of the signal slowly changed as theaverage position of the FFP was scanned along the y axis.

X-space theory was used to convert the raw signal into an image and tomake predictions regarding signal, PSF, resolution, bandwidth, andlinearity. In this section x-space theory predictions were compared withexperimental results. Two complex phantoms were imaged to demonstratethe flexibility of x-space imaging.

A. Partial Field of View Imaging

In partial field of view imaging, overlapping partial field of viewswere taken, which were reassembled into the full field of view. In FIGS.37( a), 37(b) and 37(c), an illustration of how the raw gridded signalwas baseline corrected before it was assembled into a full field of viewimage is shown. The assumption that a DC offset can approximate the lostinformation from removing the fundamental accurately enabledreconstruction.

B. PSF

X-space theory predicted the shape of the collinear PSF. By imaging aphantom smaller than the intrinsic resolution of the system, themeasured point spread function of the system was estimated. As shown inFIGS. 38( a) and 38(b), the point spread function matched thetheoretical prediction.

To calculate the theoretical point spread function, the nanoparticledistribution was determined by calculating the theoretical point spreadfunction as:

     IMG(x) = ∫_(d)fx?  ⋅ h???indicates text missing or illegible when filed

where the distribution of diameters Band f_(x) was integrated as alognormal distribution function with mean p and standard deviation σ.The signal was weighted by k because the signal was proportional to themagnetic moment m (See Eq. 1).

For the adiabatic assumption, it was assumed that the magneticnanoparticle remained aligned with the locally experienced magneticfield vector. For example, if a single magnetic nanoparticle was placedin the imager, the magnetic nanoparticle moment may always be pointingat the FFP. Introducing a single nanoparticle, the magnetic moment“flipped” to follow the FFP as the FFP passed over the nanoparticle.Since inductive detection of the signal was used, the flipping of themoment induced a signal “blip” in the receiver coil. If the movement ofthe FFP was off axis so that the FFP did not pass directly over thenanoparticle, the nanoparticle flipped slower, and so the signal “blip”was smaller and more spread out. Consequently, the PSF had the bestsignal and resolution when the FFP passed directly over thenanoparticle, and widened when the FFP no longer passed directly overthe nanoparticle.

C. Resolution

The intrinsic resolution of the system may be relevant to building aclinically relevant imaging system. As seen in the PSF image (FIGS. 38(a) and 38(b)), MPI resolution was a fundamental property of the gradientstrength and nanoparticle properties, and did not increase by increasingSNR or image reconstruction techniques.

In FIGS. 39( a) and 39(b), theoretical and measured projections werecompared across the PSF. The theoretical and measured projections showedclose correspondence, giving a 1.6 mm resolution down the bore, whichmatched the theoretically expected resolution.

FIG. 40 shows a resolution phantom that showed an image of point sourcesspaced below the resolution limit, slightly above the resolution limit,and above the resolution limit. The scan showed that the Houstoncriteria for resolution (Δx_(min)≈FWHM) was an appropriate measure forresolution.

D. Bandwidth

The reception bandwidth of the system was defined by the magnetic fieldslew rate and the properties of the magnetic nanoparticles. For a systemused with 17±3.4 nm nanoparticles, the theoretical 3 dB bandwidth wasF_(3dB)=30 kHz. Five times the theoretical bandwidth had no effect onthe width of the point spread function, which corresponded to a totalbandwidth of BW=150 kHz. The measured bandwidth compared to thetheoretical bandwidth as shown in FIG. 41. Bandwidth was measured bystepping a line source through the bore while measuring the signal. Thesignal was broken down into segments and the maximum spectral contentwas found at all frequencies up to 200 kHz. The signal at twice thefundamental frequency was reduced from the theoretical value because ouraggressive high pass filtering attenuated the signal.

In certain instances, increasing the bandwidth decreases SNR withoutimproving resolution. Harmonic number was not considered when choosingthe system bandwidth, as the number of harmonics were an artifact of arepeating excitation and not a fundamental MPI property.

E. Linearity and Shift Invariance

In certain cases, MPI is LSI when used for clinical imaging so that theimages represent both the location and quantity of magneticnanoparticles without image artifacts.

To test linearity, a linear (FIG. 42) phantom was built, which showed aline scan through a sample with linearly increasing quantities ofmagnetic tracer. The image showed both the reassembled signal and theintegral of the signal. The integral of the signal showed good linearityof the system. As can be seen, MPI is a linear imaging system whosesignal was proportional to the quantity of magnetic tracer. Theresolution phantom (FIG. 40) also doubled as a test of shift invariance.As shown in the figures, MPI was linear and shift invariant followingrecovery of the DC offset.

F. Complex Phantoms

To demonstrate the potential of MPI for imaging complex phantoms, a“CAL” phantom and an analogue of an angiography phantom were imaged.

The CAL phantom (FIGS. 43( a), 43(b) and 43(c)) showed that the systemformed a complex image that exhibited both linearity and shiftinvariance. The resolution down the bore was greater than the transverseresolution. Despite the difference the resolution between the two axes,the resulting image was clear and readable. The deconvolved image, whichwas deconvolved with the theoretical point spread function, was visuallyimproved.

The angiography phantom (FIGS. 44( a) and 44(b)) was similar in shape tothe branching of coronary arteries in the heart. The phantom was 400micron ID tubes filled with undiluted tracer embedded in raw chickentissue. Since tissue gave no MPI signal, the imager saw through thetissue without attenuation. The resulting image was simply the PSFconvolved with the original magnetization density. In some instances,MPI may be useful for angiography.

Experiment 4 A. Hardware

To test the relationship between x-space theory and bandwidth,achievable resolution and SNR, a three-dimensional MPI scanner wasbuilt. The system was constructed with permanent magnet gradient (6 T/mdown the bore and 3 T/m transverse to the bore) and an excitation coilin one dimension collinear to the bore. The FFP was rapidly scannedusing the resonant transmit coil and the signal produced was receivedwith a coil wound collinear to the transmit coil. The receive coilreceived the collinear component of the changing magnetization. Thetransmit and receive coils were collinear with the large gradient alongthe bore, which was twice the magnitude of the gradient transverse tothe bore. In some instances, the resolution in the transverse directionwas less than the resolution in the direction along the bore.

The resonant excitation coil generated 30 mT peak-to-peak at 20 kHz andwas driven by an audio amplifier (AE Techron LVC5050, Elkhart, Ind.,USA) with ˜5 kW of instantaneous power at a pulsed 2% duty cycle. Insome instances, a portion of the power of the excitation coil wasdissipated in a water cooled eddy current shield that isolated thetransmit and receive coils from the permanent magnet gradient. Thesignal from the receive coil was filtered by a passive notch filter,amplified by a FET input (e_(n)=4 nV/√{square root over (Hz)}) batterypowered preamplifier (SR560, Stanford Research Systems), and high-passfiltered at 30 kHz (SIM965, Stanford Research Systems). Following theanalog signal chain, the signal was digitized by a 16-bit dataacquisition system with a 1.25 MSPS sampling rate (National InstrumentsUSB-6259, Austin, Tex., USA), phase corrected, and low-pass filtered at400 kHz. The system was controlled by custom software written in MATLAB(Mathworks MATLAB, Natick, Mass., USA).

B. Imaging

The 30 mT peak-to-peak excitation enabled a partial FOV of approximately0.5 cm along the z-axis. The signal for the received partial FOV wasscaled to the instantaneous speed of the FFP, gridded to theinstantaneous location of the FFP, and thus assigned to a physicallocation on the phantom. The phantom was stepped in 100 μm to 2.5 mmincrements along the z-axis, acquiring a partial FOV line scan at eachstep.

Following data acquisition, the raw data was gridded using x-spacetheory to form a partial line scan. The partial line scans werereassembled by estimating the missing DC offset from scan to scan togenerate an assembled full FOV of up to 8 cm along the z-axis.

Phantoms were constructed using 400 micron ID tubing filled withundiluted SPIO Resovist tracer (Bayer-Schering). The resolution phantomwas constructed using three sets of tubing separated by 1 mm, 2 mm, and3 mm, respectively. The intrinsic resolution of the system wasapproximately 1.6 mm, so this resulted in one set of points below theresolvability limit, slightly above the limit, and solidly above thelimit.

C. Results and Discussion

In FIGS. 48( a), 48(b) and 48(c), the experimental signal power is shownas a function of bandwidth compared with the theoretical bandwidthpower. Bandwidth usage was calculated by moving a point source down thebore in 200 micron increments and taking a spectrum at each position,ensuring that the phantom passed through the center of the bore. Animage of the raw frequency data is shown in FIG. 48( a). The signalsummed across all the point source locations is shown in FIG. 48( c).The signal power in decibels is shown in FIG. 48( b).

FIG. 49( a) shows the one-dimensional MPI images of a resolutionphantom. Prior to assembly, the partial field of view scans werebaseline corrected so that they maximized their overlap. An example ofunassembled data following the scaling and gridding step is shown inFIG. 49( b). The baseline corrected image was assembled into aone-dimensional image as seen in FIG. 49( c).

In FIG. 50 shows the measured resolution of the system as a function ofthe input bandwidth without any deconvolution. The measured FWHM of 2 mmwas wider than the system resolution because the point source had adiameter of 400 μm. In some cases, the FWHM asymptotically improved withincreasing bandwidth.

In FIG. 51, the measured noise is shown as a function of the inputbandwidth. The measured RMS noise increased with the system bandwidth.Continuing to increase the bandwidth beyond what was required for atarget resolution decreased SNR and did not substantially increaseresolution.

The optimal imaging bandwidth in x-space MPI required a tradeoff betweenachievable resolution and SNR. Like all Linear and Shift Invariant (LSI)imaging systems, the intrinsic resolution of MPI was not related tosystem SNR, and was instead a property of how the magnetic nanoparticleinteracts with the gradient. However, the intrinsic resolution wasdegraded by the choice of a low pass filter, as shown in FIG. 50. Insome instances, the noise increased with the bandwidth (see FIG. 51).

These results indicated that, when imaging Resovist magnetic particles,the signal bandwidth did not need to increase above 200 kHz for a 30 mTpeak-peak magnetic field at 20 kHz. Continuing to increase the bandwidthdid not increase the measured resolution appreciably, and increased thesystem noise. The value of this required bandwidth changed with theproperties of the particle and the magnetic field slew rate. In certainembodiments, the system bandwidth may be reduced, decreasing resolutionin order to improve SNR at the expense of resolution. For example,halving the receive bandwidth to 100 kHz theoretically improved SNR bymore than forty percent, while decreasing the measured resolution bytwenty percent. The experimental results indicated that ≈1.6 mm was theresolution of Resovist tracer in an MPI system with a 6 T/m gradient(FIG. 50).

The models indicated that using the root mean squared (RMS) magneticfield slew rate accurately modeled the system bandwidth requirements.While the peak FFP movement speed may be faster than the RMS speed, theFFP movement speed was averaged across the FOV.

Various other modifications and alternations in the structure and methodof operation of the present disclosure will be apparent to those skilledin the art without departing from the scope and spirit of the presentdisclosure. Although the present disclosure has been described inconnection with specific preferred embodiments, it should be understoodthat the present disclosure as claimed should not be unduly limited tosuch specific embodiments.

1. A magnetic particle imaging device comprising: a magnetic fieldsource configured to produce a magnetic field having a non-saturatingmagnetic field region; an excitation signal source configured to producean excitation signal in the non-saturating magnetic field region thatproduces a detectable signal from magnetic particles in thenon-saturating magnetic field region; and a signal processor configuredto convert the detectable signal into an image of the magneticparticles.
 2. The device of claim 1, wherein the image is aone-dimensional image, a two-dimensional image, or a three-dimensionalimage.
 3. The device of claim 1, wherein the image includes a pluralityof images of the magnetic particles in the sample over a period of time.4. The device of claim 1, wherein the magnetic field source comprisestwo or more coaxially arranged magnetic field sources.
 5. The device ofclaim 4, wherein the magnetic field sources are permanent magnets. 6.The device of claim 4, wherein the magnetic field has a magnetic fieldgradient ranging from 0.5 Tesla/meter to 30 Tesla/meter.
 7. The deviceof claim 1, wherein the excitation signal comprises a RF excitationsignal.
 8. The device of claim 7, wherein the excitation signal furthercomprises an intermodulation excitation signal.
 9. The device of claim1, wherein the device further comprises a scanning magnetic field sourceconfigured to produce a scanning magnetic field that positions thenon-saturating magnetic field region in the magnetic field.
 10. Thedevice of claim 9, wherein the device is configured to produce alinearly varying signal with respect to the concentration of magneticparticles in the non-saturating magnetic field region.
 11. The device ofclaim 1, wherein the device further comprises a receiver configured todetect the signal from the magnetic particles in the non-saturatingmagnetic field region.
 12. The device of claim 11, wherein the receivercomprises a receiver coil with a Q factor of 100 or more.
 13. The deviceof claim 11, wherein the receiver is configured to have a receivebandwidth ranging from 10 kHz to 1 MHz.
 14. The device of claim 1,wherein the device is configured to have a signal to noise ratio rangingfrom 10 to 500,000.
 15. The device of claim 1, wherein the device isconfigured to have a resolution ranging from 1 mm to 100 μm.
 16. Thedevice of claim 1, wherein the device is configured to have asensitivity ranging from 1 μg to 0.1 ng.
 17. The device of claim 1,wherein the device is configured to have a field of view ranging from 1cm to 50 cm.
 18. A method of producing an image of magnetic particles ina sample, the method comprising: applying a magnetic field having anon-saturating magnetic field region to a sample comprising magneticparticles; applying an excitation signal to the magnetic particles inthe non-saturating magnetic field region to produce a detectable signalfrom the magnetic particles in the non-saturating magnetic field region;detecting the signal from the magnetic particles in the non-saturatingmagnetic field region; and analyzing the detected signal to produce animage of the magnetic particles in the sample.
 19. The method of claim18, further comprising applying a scanning magnetic field to themagnetic field having the non-saturating magnetic field region toposition the non-saturating magnetic field region in the magnetic field.20. The method of claim 19, further comprising: repositioning thenon-saturating magnetic field region in the magnetic field; andrepeating the detecting and repositioning to detect two or more signalsfrom the magnetic particles in the non-saturating magnetic field region.21. The method of claim 20, wherein the analyzing comprises correlatingthe detected signal to the position of the non-saturating magnetic fieldregion when the signal was detected.
 22. The method of claim 20, whereinthe analyzing comprises converting the detected signals into partialfield of view images.
 23. The method of claim 22, wherein the analyzingfurther comprises combining the partial field of view images to producethe image of the magnetic particles in the sample.
 24. The method ofclaim 19, further comprising varying the magnetic field strength of themagnetic field while applying the scanning magnetic field to produceimages having different resolutions.
 25. The method of claim 24, furthercomprising analyzing the images having different resolutions todetermine low frequency image data.
 26. The method of claim 18, whereinthe applying the excitation signal comprises applying an RF excitationsignal to the magnetic particles in the non-saturating magnetic fieldregion.
 27. The method of claim 26, wherein the applying the excitationsignal further comprises applying an intermodulation signal to themagnetic particles in the non-saturating magnetic field region.
 28. Themethod of claim 18, wherein the method comprises producing a pluralityof images of magnetic particles in the sample over a period of time. 29.A method of producing an image of magnetic particles in a subject, themethod comprising: administering magnetic particles to a subject;positioning the subject in a magnetic particle imaging device; applyinga magnetic field having a non-saturating magnetic field region to thesubject; applying an excitation signal to the magnetic particles in thenon-saturating magnetic field region to produce a detectable signal fromthe magnetic particles in the non-saturating magnetic field region;detecting the signal from the magnetic particles in the non-saturatingmagnetic field region; and analyzing the detected signal to produce animage of the magnetic particles in the subject.